%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : COM203^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n024.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:41:56 AM UTC 2026
% Result : Theorem 3.16s 1.74s
% Output : Refutation 3.16s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : COM203^1 : TPTP v9.2.1. Released v7.0.0.
% 0.00/0.09 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.10/0.29 % Computer : n024.cluster.edu
% 0.10/0.29 % Model : x86_64 x86_64
% 0.10/0.29 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.29 % Memory : 8042.1875MB
% 0.10/0.29 % OS : Linux 3.10.0-693.el7.x86_64
% 0.10/0.29 % CPULimit : 300
% 0.10/0.29 % WCLimit : 300
% 0.10/0.29 % DateTime : Mon Feb 23 23:16:40 EST 2026
% 0.10/0.29 % CPUTime :
% 0.10/0.29 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.38 ---- Original DTF file ---
% 0.17/0.38 thf(spec,logic,$$dhol).
% 0.17/0.38 %------------------------------------------------------------------------------
% 0.17/0.38 % File : COM203^1 : TPTP v9.2.1. Released v7.0.0.
% 0.17/0.38 % Domain : Computing Theory
% 0.17/0.38 % Problem : Parallel extension to grammars and languages 44
% 0.17/0.38 % Version : [Bla16] axioms : Especial.
% 0.17/0.38 % English :
% 0.17/0.38
% 0.17/0.38 % Refs : [BH+14] Blanchette et al. (2014), Truly Modular (Co)datatypes
% 0.17/0.38 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.17/0.38 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.17/0.38 % Source : [Bla16]
% 0.17/0.38 % Names : parallel__44.p [Bla16]
% 0.17/0.38
% 0.17/0.38 % Status : Theorem
% 0.17/0.38 % Rating : 0.33 v9.1.0, 0.00 v7.2.0, 0.25 v7.1.0
% 0.17/0.38 % Syntax : Number of formulae : 329 ( 123 unt; 57 typ; 0 def)
% 0.17/0.38 % Number of atoms : 592 ( 250 equ; 0 cnn)
% 0.17/0.38 % Maximal formula atoms : 7 ( 2 avg)
% 0.17/0.38 % Number of connectives : 4381 ( 46 ~; 2 |; 27 &;4031 @)
% 0.17/0.38 % ( 0 <=>; 275 =>; 0 <=; 0 <~>)
% 0.17/0.38 % Maximal formula depth : 28 ( 9 avg)
% 0.17/0.38 % Number of types : 4 ( 3 usr)
% 0.17/0.38 % Number of type conns : 363 ( 363 >; 0 *; 0 +; 0 <<)
% 0.17/0.38 % Number of symbols : 56 ( 54 usr; 3 con; 0-7 aty)
% 0.17/0.38 % Number of variables : 1272 ( 44 ^;1137 !; 18 ?;1272 :)
% 0.17/0.38 % ( 73 !>; 0 ?*; 0 @-; 0 @+)
% 0.17/0.38 % SPC : TH1_THM_EQU_NAR
% 0.17/0.38
% 0.17/0.38 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.17/0.38 % 2016-07-13 14:38:49.144
% 0.17/0.38 %------------------------------------------------------------------------------
% 0.17/0.38 %----Could-be-implicit typings (8)
% 0.17/0.38 thf(ty_t_Product__Type_Oprod,type,
% 0.17/0.38 product_prod: $tType > $tType > $tType ).
% 0.17/0.38
% 0.17/0.38 thf(ty_t_Sum__Type_Osum,type,
% 0.17/0.38 sum_sum: $tType > $tType > $tType ).
% 0.17/0.38
% 0.17/0.38 thf(ty_t_Option_Ooption,type,
% 0.17/0.38 option: $tType > $tType ).
% 0.17/0.38
% 0.17/0.38 thf(ty_t_DTree_Odtree,type,
% 0.17/0.38 dtree: $tType ).
% 0.17/0.38
% 0.17/0.38 thf(ty_t_Set_Oset,type,
% 0.17/0.38 set: $tType > $tType ).
% 0.17/0.38
% 0.17/0.38 thf(ty_t_DTree_OT,type,
% 0.17/0.38 t: $tType ).
% 0.17/0.38
% 0.17/0.38 thf(ty_t_DTree_ON,type,
% 0.17/0.38 n: $tType ).
% 0.17/0.38
% 0.17/0.38 thf(ty_t_itself,type,
% 0.17/0.38 itself: $tType > $tType ).
% 0.17/0.38
% 0.17/0.38 %----Explicit typings (49)
% 0.17/0.38 thf(sy_cl_HOL_Otype,type,
% 0.17/0.38 type:
% 0.17/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_cl_Orderings_Oord,type,
% 0.17/0.38 ord:
% 0.17/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_cl_Orderings_Otop,type,
% 0.17/0.38 top:
% 0.17/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.17/0.38 finite_finite:
% 0.17/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_cl_Nat_Osemiring__char__0,type,
% 0.17/0.38 semiring_char_0:
% 0.17/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_cl_Orderings_Oorder__top,type,
% 0.17/0.38 order_top:
% 0.17/0.38 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 0.17/0.38 bNF_Ca1785829860lChain:
% 0.17/0.38 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_BNF__Def_OfstOp,type,
% 0.17/0.38 bNF_fstOp:
% 0.17/0.38 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > ( product_prod @ A @ C ) > ( product_prod @ A @ B ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_BNF__Def_Opick__middlep,type,
% 0.17/0.38 bNF_pick_middlep:
% 0.17/0.38 !>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > A ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_BNF__Def_OsndOp,type,
% 0.17/0.38 bNF_sndOp:
% 0.17/0.38 !>[C: $tType,A: $tType,B: $tType] : ( ( C > A > $o ) > ( A > B > $o ) > ( product_prod @ C @ B ) > ( product_prod @ A @ B ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
% 0.17/0.38 bNF_Greatest_image2:
% 0.17/0.38 !>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_DTree_ONode,type,
% 0.17/0.38 node: n > ( set @ ( sum_sum @ t @ dtree ) ) > dtree ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_DTree_Ocont,type,
% 0.17/0.38 cont: dtree > ( set @ ( sum_sum @ t @ dtree ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_DTree_Ocorec,type,
% 0.17/0.38 corec:
% 0.17/0.38 !>[A: $tType] : ( ( A > n ) > ( A > ( set @ ( sum_sum @ t @ ( sum_sum @ dtree @ A ) ) ) ) > A > dtree ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_DTree_Odtree_Oroot,type,
% 0.17/0.38 root: dtree > n ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_DTree_Ounfold,type,
% 0.17/0.38 unfold:
% 0.17/0.38 !>[A: $tType] : ( ( A > n ) > ( A > ( set @ ( sum_sum @ t @ A ) ) ) > A > dtree ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_Finite__Set_Ofinite,type,
% 0.17/0.38 finite_finite2:
% 0.17/0.38 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_Fun_Oinj__on,type,
% 0.17/0.38 inj_on:
% 0.17/0.38 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_Fun_Oswap,type,
% 0.17/0.38 swap:
% 0.17/0.38 !>[A: $tType,B: $tType] : ( A > A > ( A > B ) > A > B ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 0.17/0.38 inf_inf:
% 0.17/0.38 !>[A: $tType] : ( A > A > A ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.17/0.38 ord_less_eq:
% 0.17/0.38 !>[A: $tType] : ( A > A > $o ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.17/0.38 top_top:
% 0.17/0.38 !>[A: $tType] : A ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_ONplus,type,
% 0.17/0.38 parall1518086719_Nplus: n > n > n ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar,type,
% 0.17/0.38 parall1899940088le_par: ( product_prod @ dtree @ dtree ) > dtree ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__c,type,
% 0.17/0.38 parall1914194347_par_c: ( product_prod @ dtree @ dtree ) > ( set @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__r,type,
% 0.17/0.38 parall1914194362_par_r: ( product_prod @ dtree @ dtree ) > n ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__r__rel,type,
% 0.17/0.38 parall556292031_r_rel: ( product_prod @ dtree @ dtree ) > ( product_prod @ dtree @ dtree ) > $o ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_Product__Type_OPair,type,
% 0.17/0.38 product_Pair:
% 0.17/0.38 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_Product__Type_Ocurry,type,
% 0.17/0.38 product_curry:
% 0.17/0.38 !>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.17/0.38 produc2004651681e_prod:
% 0.17/0.38 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_Product__Type_Omap__prod,type,
% 0.17/0.38 product_map_prod:
% 0.17/0.38 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.17/0.38 product_rec_prod:
% 0.17/0.38 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 0.17/0.38 product_case_prod:
% 0.17/0.38 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 0.17/0.38 product_fst:
% 0.17/0.38 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 0.17/0.38 product_snd:
% 0.17/0.38 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 0.17/0.38 product_swap:
% 0.17/0.38 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_Product__Type_Oscomp,type,
% 0.17/0.38 product_scomp:
% 0.17/0.38 !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_Pure_Otype,type,
% 0.17/0.38 type2:
% 0.17/0.38 !>[A: $tType] : ( itself @ A ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_Relation_Oinv__image,type,
% 0.17/0.38 inv_image:
% 0.17/0.38 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_Set_OCollect,type,
% 0.17/0.38 collect:
% 0.17/0.38 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_Set_OPow,type,
% 0.17/0.38 pow:
% 0.17/0.38 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_Set_Oimage,type,
% 0.17/0.38 image:
% 0.17/0.38 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_Set_Ovimage,type,
% 0.17/0.38 vimage:
% 0.17/0.38 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ B ) > ( set @ A ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_Wellfounded_Oaccp,type,
% 0.17/0.38 accp:
% 0.17/0.38 !>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_Wellfounded_Olex__prod,type,
% 0.17/0.38 lex_prod:
% 0.17/0.38 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_Wfrec_Osame__fst,type,
% 0.17/0.38 same_fst:
% 0.17/0.38 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_c_member,type,
% 0.17/0.38 member:
% 0.17/0.38 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.17/0.38
% 0.17/0.38 thf(sy_v_tr1,type,
% 0.17/0.38 tr1: dtree ).
% 0.17/0.38
% 0.17/0.38 thf(sy_v_tr2,type,
% 0.17/0.38 tr2: dtree ).
% 0.17/0.38
% 0.17/0.38 %----Relevant facts (256)
% 0.17/0.38 thf(fact_0__092_060open_062root_A_Iunfold_Apar__r_Apar__c_A_Itr1_M_Atr2_J_J_A_061_Apar__r_A_Itr1_M_Atr2_J_092_060close_062,axiom,
% 0.17/0.38 ( ( root @ ( unfold @ ( product_prod @ dtree @ dtree ) @ parall1914194362_par_r @ parall1914194347_par_c @ ( product_Pair @ dtree @ dtree @ tr1 @ tr2 ) ) )
% 0.17/0.38 = ( parall1914194362_par_r @ ( product_Pair @ dtree @ dtree @ tr1 @ tr2 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % \<open>root (unfold par_r par_c (tr1, tr2)) = par_r (tr1, tr2)\<close>
% 0.17/0.38 thf(fact_1_Nplus__comm,axiom,
% 0.17/0.38 ( parall1518086719_Nplus
% 0.17/0.38 = ( ^ [A2: n,B2: n] : ( parall1518086719_Nplus @ B2 @ A2 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % Nplus_comm
% 0.17/0.38 thf(fact_2_Nplus__assoc,axiom,
% 0.17/0.38 ! [A3: n,B3: n,C2: n] :
% 0.17/0.38 ( ( parall1518086719_Nplus @ ( parall1518086719_Nplus @ A3 @ B3 ) @ C2 )
% 0.17/0.38 = ( parall1518086719_Nplus @ A3 @ ( parall1518086719_Nplus @ B3 @ C2 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % Nplus_assoc
% 0.17/0.38 thf(fact_3_par__r_Ocases,axiom,
% 0.17/0.38 ! [X: product_prod @ dtree @ dtree] :
% 0.17/0.38 ~ ! [Tr1: dtree,Tr2: dtree] :
% 0.17/0.38 ( X
% 0.17/0.38 != ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) ) ).
% 0.17/0.38
% 0.17/0.38 % par_r.cases
% 0.17/0.38 thf(fact_4_par__r_Oelims,axiom,
% 0.17/0.38 ! [X: product_prod @ dtree @ dtree,Y: n] :
% 0.17/0.38 ( ( ( parall1914194362_par_r @ X )
% 0.17/0.38 = Y )
% 0.17/0.38 => ~ ! [Tr1: dtree,Tr2: dtree] :
% 0.17/0.38 ( ( X
% 0.17/0.38 = ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) )
% 0.17/0.38 => ( Y
% 0.17/0.38 != ( parall1518086719_Nplus @ ( root @ Tr1 ) @ ( root @ Tr2 ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % par_r.elims
% 0.17/0.38 thf(fact_5_par__r_Osimps,axiom,
% 0.17/0.38 ! [Tr12: dtree,Tr22: dtree] :
% 0.17/0.38 ( ( parall1914194362_par_r @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) )
% 0.17/0.38 = ( parall1518086719_Nplus @ ( root @ Tr12 ) @ ( root @ Tr22 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % par_r.simps
% 0.17/0.38 thf(fact_6_par__r_Oinduct,axiom,
% 0.17/0.38 ! [P: ( product_prod @ dtree @ dtree ) > $o,A0: product_prod @ dtree @ dtree] :
% 0.17/0.38 ( ! [Tr1: dtree,Tr2: dtree] : ( P @ ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) )
% 0.17/0.38 => ( P @ A0 ) ) ).
% 0.17/0.38
% 0.17/0.38 % par_r.induct
% 0.17/0.38 thf(fact_7_par__def,axiom,
% 0.17/0.38 ( parall1899940088le_par
% 0.17/0.38 = ( unfold @ ( product_prod @ dtree @ dtree ) @ parall1914194362_par_r @ parall1914194347_par_c ) ) ).
% 0.17/0.38
% 0.17/0.38 % par_def
% 0.17/0.38 thf(fact_8_unfold_I1_J,axiom,
% 0.17/0.38 ! [A: $tType,Rt: A > n,Ct: A > ( set @ ( sum_sum @ t @ A ) ),B3: A] :
% 0.17/0.38 ( ( root @ ( unfold @ A @ Rt @ Ct @ B3 ) )
% 0.17/0.38 = ( Rt @ B3 ) ) ).
% 0.17/0.38
% 0.17/0.38 % unfold(1)
% 0.17/0.38 thf(fact_9_prod_Oinject,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
% 0.17/0.38 ( ( ( product_Pair @ A @ B @ X1 @ X2 )
% 0.17/0.38 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 0.17/0.38 = ( ( X1 = Y1 )
% 0.17/0.38 & ( X2 = Y2 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % prod.inject
% 0.17/0.38 thf(fact_10_old_Oprod_Oinject,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
% 0.17/0.38 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 0.17/0.38 = ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.17/0.38 = ( ( A3 = A4 )
% 0.17/0.38 & ( B3 = B4 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % old.prod.inject
% 0.17/0.38 thf(fact_11_par__r_Opelims,axiom,
% 0.17/0.38 ! [X: product_prod @ dtree @ dtree,Y: n] :
% 0.17/0.38 ( ( ( parall1914194362_par_r @ X )
% 0.17/0.38 = Y )
% 0.17/0.38 => ( ( accp @ ( product_prod @ dtree @ dtree ) @ parall556292031_r_rel @ X )
% 0.17/0.38 => ~ ! [Tr1: dtree,Tr2: dtree] :
% 0.17/0.38 ( ( X
% 0.17/0.38 = ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) )
% 0.17/0.38 => ( ( Y
% 0.17/0.38 = ( parall1518086719_Nplus @ ( root @ Tr1 ) @ ( root @ Tr2 ) ) )
% 0.17/0.38 => ~ ( accp @ ( product_prod @ dtree @ dtree ) @ parall556292031_r_rel @ ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % par_r.pelims
% 0.17/0.38 thf(fact_12_surj__pair,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 0.17/0.38 ? [X3: A,Y3: B] :
% 0.17/0.38 ( P2
% 0.17/0.38 = ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% 0.17/0.38
% 0.17/0.38 % surj_pair
% 0.17/0.38 thf(fact_13_prod__cases,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 0.17/0.38 ( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.17/0.38 => ( P @ P2 ) ) ).
% 0.17/0.38
% 0.17/0.38 % prod_cases
% 0.17/0.38 thf(fact_14_Pair__inject,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
% 0.17/0.38 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 0.17/0.38 = ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.17/0.38 => ~ ( ( A3 = A4 )
% 0.17/0.38 => ( B3 != B4 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % Pair_inject
% 0.17/0.38 thf(fact_15_prod__cases3,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.17/0.38 ~ ! [A5: A,B5: B,C3: C] :
% 0.17/0.38 ( Y
% 0.17/0.38 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % prod_cases3
% 0.17/0.38 thf(fact_16_prod__cases4,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.17/0.38 ~ ! [A5: A,B5: B,C3: C,D2: D] :
% 0.17/0.38 ( Y
% 0.17/0.38 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % prod_cases4
% 0.17/0.38 thf(fact_17_prod__cases5,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.17/0.38 ~ ! [A5: A,B5: B,C3: C,D2: D,E2: E] :
% 0.17/0.38 ( Y
% 0.17/0.38 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % prod_cases5
% 0.17/0.38 thf(fact_18_prod__cases6,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 0.17/0.38 ~ ! [A5: A,B5: B,C3: C,D2: D,E2: E,F2: F] :
% 0.17/0.38 ( Y
% 0.17/0.38 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % prod_cases6
% 0.17/0.38 thf(fact_19_old_Oprod_Oinducts,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.17/0.38 ( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.17/0.38 => ( P @ Prod ) ) ).
% 0.17/0.38
% 0.17/0.38 % old.prod.inducts
% 0.17/0.38 thf(fact_20_old_Oprod_Oexhaust,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 0.17/0.38 ~ ! [A5: A,B5: B] :
% 0.17/0.38 ( Y
% 0.17/0.38 != ( product_Pair @ A @ B @ A5 @ B5 ) ) ).
% 0.17/0.38
% 0.17/0.38 % old.prod.exhaust
% 0.17/0.38 thf(fact_21_prod__induct7,axiom,
% 0.17/0.38 ! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 0.17/0.38 ( ! [A5: A,B5: B,C3: C,D2: D,E2: E,F2: F,G2: G] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
% 0.17/0.38 => ( P @ X ) ) ).
% 0.17/0.38
% 0.17/0.38 % prod_induct7
% 0.17/0.38 thf(fact_22_prod__induct6,axiom,
% 0.17/0.38 ! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 0.17/0.38 ( ! [A5: A,B5: B,C3: C,D2: D,E2: E,F2: F] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
% 0.17/0.38 => ( P @ X ) ) ).
% 0.17/0.38
% 0.17/0.38 % prod_induct6
% 0.17/0.38 thf(fact_23_prod__induct5,axiom,
% 0.17/0.38 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.17/0.38 ( ! [A5: A,B5: B,C3: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 0.17/0.38 => ( P @ X ) ) ).
% 0.17/0.38
% 0.17/0.38 % prod_induct5
% 0.17/0.38 thf(fact_24_prod__induct4,axiom,
% 0.17/0.38 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.17/0.38 ( ! [A5: A,B5: B,C3: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
% 0.17/0.38 => ( P @ X ) ) ).
% 0.17/0.38
% 0.17/0.38 % prod_induct4
% 0.17/0.38 thf(fact_25_prod__induct3,axiom,
% 0.17/0.38 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.17/0.38 ( ! [A5: A,B5: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) )
% 0.17/0.38 => ( P @ X ) ) ).
% 0.17/0.38
% 0.17/0.38 % prod_induct3
% 0.17/0.38 thf(fact_26_prod__cases7,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 0.17/0.38 ~ ! [A5: A,B5: B,C3: C,D2: D,E2: E,F2: F,G2: G] :
% 0.17/0.38 ( Y
% 0.17/0.38 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % prod_cases7
% 0.17/0.38 thf(fact_27_old_Oprod_Orec,axiom,
% 0.17/0.38 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B3: B] :
% 0.17/0.38 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.17/0.38 = ( F1 @ A3 @ B3 ) ) ).
% 0.17/0.38
% 0.17/0.38 % old.prod.rec
% 0.17/0.38 thf(fact_28_internal__case__prod__conv,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A3: B,B3: C] :
% 0.17/0.38 ( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A3 @ B3 ) )
% 0.17/0.38 = ( C2 @ A3 @ B3 ) ) ).
% 0.17/0.38
% 0.17/0.38 % internal_case_prod_conv
% 0.17/0.38 thf(fact_29_root__Node,axiom,
% 0.17/0.38 ! [N: n,As: set @ ( sum_sum @ t @ dtree )] :
% 0.17/0.38 ( ( root @ ( node @ N @ As ) )
% 0.17/0.38 = N ) ).
% 0.17/0.38
% 0.17/0.38 % root_Node
% 0.17/0.38 thf(fact_30_accp_Ocases,axiom,
% 0.17/0.38 ! [A: $tType,R: A > A > $o,A3: A] :
% 0.17/0.38 ( ( accp @ A @ R @ A3 )
% 0.17/0.38 => ! [Y4: A] :
% 0.17/0.38 ( ( R @ Y4 @ A3 )
% 0.17/0.38 => ( accp @ A @ R @ Y4 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % accp.cases
% 0.17/0.38 thf(fact_31_accp_Osimps,axiom,
% 0.17/0.38 ! [A: $tType] :
% 0.17/0.38 ( ( accp @ A )
% 0.17/0.38 = ( ^ [R2: A > A > $o,A2: A] :
% 0.17/0.38 ? [X4: A] :
% 0.17/0.38 ( ( A2 = X4 )
% 0.17/0.38 & ! [Y5: A] :
% 0.17/0.38 ( ( R2 @ Y5 @ X4 )
% 0.17/0.38 => ( accp @ A @ R2 @ Y5 ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % accp.simps
% 0.17/0.38 thf(fact_32_accp_Ointros,axiom,
% 0.17/0.38 ! [A: $tType,R: A > A > $o,X: A] :
% 0.17/0.38 ( ! [Y3: A] :
% 0.17/0.38 ( ( R @ Y3 @ X )
% 0.17/0.38 => ( accp @ A @ R @ Y3 ) )
% 0.17/0.38 => ( accp @ A @ R @ X ) ) ).
% 0.17/0.38
% 0.17/0.38 % accp.intros
% 0.17/0.38 thf(fact_33_accp__induct,axiom,
% 0.17/0.38 ! [A: $tType,R: A > A > $o,A3: A,P: A > $o] :
% 0.17/0.38 ( ( accp @ A @ R @ A3 )
% 0.17/0.38 => ( ! [X3: A] :
% 0.17/0.38 ( ( accp @ A @ R @ X3 )
% 0.17/0.38 => ( ! [Y4: A] :
% 0.17/0.38 ( ( R @ Y4 @ X3 )
% 0.17/0.38 => ( P @ Y4 ) )
% 0.17/0.38 => ( P @ X3 ) ) )
% 0.17/0.38 => ( P @ A3 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % accp_induct
% 0.17/0.38 thf(fact_34_accp_Oinducts,axiom,
% 0.17/0.38 ! [A: $tType,R: A > A > $o,X: A,P: A > $o] :
% 0.17/0.38 ( ( accp @ A @ R @ X )
% 0.17/0.38 => ( ! [X3: A] :
% 0.17/0.38 ( ! [Y4: A] :
% 0.17/0.38 ( ( R @ Y4 @ X3 )
% 0.17/0.38 => ( accp @ A @ R @ Y4 ) )
% 0.17/0.38 => ( ! [Y4: A] :
% 0.17/0.38 ( ( R @ Y4 @ X3 )
% 0.17/0.38 => ( P @ Y4 ) )
% 0.17/0.38 => ( P @ X3 ) ) )
% 0.17/0.38 => ( P @ X ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % accp.inducts
% 0.17/0.38 thf(fact_35_accp__downward,axiom,
% 0.17/0.38 ! [A: $tType,R: A > A > $o,B3: A,A3: A] :
% 0.17/0.38 ( ( accp @ A @ R @ B3 )
% 0.17/0.38 => ( ( R @ A3 @ B3 )
% 0.17/0.38 => ( accp @ A @ R @ A3 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % accp_downward
% 0.17/0.38 thf(fact_36_not__accp__down,axiom,
% 0.17/0.38 ! [A: $tType,R3: A > A > $o,X: A] :
% 0.17/0.38 ( ~ ( accp @ A @ R3 @ X )
% 0.17/0.38 => ~ ! [Z: A] :
% 0.17/0.38 ( ( R3 @ Z @ X )
% 0.17/0.38 => ( accp @ A @ R3 @ Z ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % not_accp_down
% 0.17/0.38 thf(fact_37_accp__induct__rule,axiom,
% 0.17/0.38 ! [A: $tType,R: A > A > $o,A3: A,P: A > $o] :
% 0.17/0.38 ( ( accp @ A @ R @ A3 )
% 0.17/0.38 => ( ! [X3: A] :
% 0.17/0.38 ( ( accp @ A @ R @ X3 )
% 0.17/0.38 => ( ! [Y4: A] :
% 0.17/0.38 ( ( R @ Y4 @ X3 )
% 0.17/0.38 => ( P @ Y4 ) )
% 0.17/0.38 => ( P @ X3 ) ) )
% 0.17/0.38 => ( P @ A3 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % accp_induct_rule
% 0.17/0.38 thf(fact_38_in__lex__prod,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] :
% 0.17/0.38 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Pair @ A @ B @ A4 @ B4 ) ) @ ( lex_prod @ A @ B @ R @ S ) )
% 0.17/0.38 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A4 ) @ R )
% 0.17/0.38 | ( ( A3 = A4 )
% 0.17/0.38 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B3 @ B4 ) @ S ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % in_lex_prod
% 0.17/0.38 thf(fact_39_Node__root__cont,axiom,
% 0.17/0.38 ! [Tr: dtree] :
% 0.17/0.38 ( ( node @ ( root @ Tr ) @ ( cont @ Tr ) )
% 0.17/0.38 = Tr ) ).
% 0.17/0.38
% 0.17/0.38 % Node_root_cont
% 0.17/0.38 thf(fact_40_corec_I1_J,axiom,
% 0.17/0.38 ! [A: $tType,Rt: A > n,Ct: A > ( set @ ( sum_sum @ t @ ( sum_sum @ dtree @ A ) ) ),B3: A] :
% 0.17/0.38 ( ( root @ ( corec @ A @ Rt @ Ct @ B3 ) )
% 0.17/0.38 = ( Rt @ B3 ) ) ).
% 0.17/0.38
% 0.17/0.38 % corec(1)
% 0.17/0.38 thf(fact_41_ssubst__Pair__rhs,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,R: A,S: B,R3: set @ ( product_prod @ A @ B ),S2: B] :
% 0.17/0.38 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S ) @ R3 )
% 0.17/0.38 => ( ( S2 = S )
% 0.17/0.38 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S2 ) @ R3 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % ssubst_Pair_rhs
% 0.17/0.38 thf(fact_42_curry__conv,axiom,
% 0.17/0.38 ! [A: $tType,C: $tType,B: $tType] :
% 0.17/0.38 ( ( product_curry @ B @ C @ A )
% 0.17/0.38 = ( ^ [F3: ( product_prod @ B @ C ) > A,A2: B,B2: C] : ( F3 @ ( product_Pair @ B @ C @ A2 @ B2 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % curry_conv
% 0.17/0.38 thf(fact_43_finite__par__c,axiom,
% 0.17/0.38 ! [Tr12: dtree,Tr22: dtree] : ( finite_finite2 @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) @ ( parall1914194347_par_c @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % finite_par_c
% 0.17/0.38 thf(fact_44_curryI,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,F4: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
% 0.17/0.38 ( ( F4 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.17/0.38 => ( product_curry @ A @ B @ $o @ F4 @ A3 @ B3 ) ) ).
% 0.17/0.38
% 0.17/0.38 % curryI
% 0.17/0.38 thf(fact_45_mem__Collect__eq,axiom,
% 0.17/0.38 ! [A: $tType,A3: A,P: A > $o] :
% 0.17/0.38 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 0.17/0.38 = ( P @ A3 ) ) ).
% 0.17/0.38
% 0.17/0.38 % mem_Collect_eq
% 0.17/0.38 thf(fact_46_Collect__mem__eq,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A] :
% 0.17/0.38 ( ( collect @ A
% 0.17/0.38 @ ^ [X4: A] : ( member @ A @ X4 @ A6 ) )
% 0.17/0.38 = A6 ) ).
% 0.17/0.38
% 0.17/0.38 % Collect_mem_eq
% 0.17/0.38 thf(fact_47_Collect__cong,axiom,
% 0.17/0.38 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.17/0.38 ( ! [X3: A] :
% 0.17/0.38 ( ( P @ X3 )
% 0.17/0.38 = ( Q @ X3 ) )
% 0.17/0.38 => ( ( collect @ A @ P )
% 0.17/0.38 = ( collect @ A @ Q ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % Collect_cong
% 0.17/0.38 thf(fact_48_ext,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,F4: A > B,G3: A > B] :
% 0.17/0.38 ( ! [X3: A] :
% 0.17/0.38 ( ( F4 @ X3 )
% 0.17/0.38 = ( G3 @ X3 ) )
% 0.17/0.38 => ( F4 = G3 ) ) ).
% 0.17/0.38
% 0.17/0.38 % ext
% 0.17/0.38 thf(fact_49_swap__simp,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,X: B,Y: A] :
% 0.17/0.38 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 0.17/0.38 = ( product_Pair @ A @ B @ Y @ X ) ) ).
% 0.17/0.38
% 0.17/0.38 % swap_simp
% 0.17/0.38 thf(fact_50_internal__case__prod__def,axiom,
% 0.17/0.38 ! [C: $tType,B: $tType,A: $tType] :
% 0.17/0.38 ( ( produc2004651681e_prod @ A @ B @ C )
% 0.17/0.38 = ( product_case_prod @ A @ B @ C ) ) ).
% 0.17/0.38
% 0.17/0.38 % internal_case_prod_def
% 0.17/0.38 thf(fact_51_swap__swap,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 0.17/0.38 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
% 0.17/0.38 = P2 ) ).
% 0.17/0.38
% 0.17/0.38 % swap_swap
% 0.17/0.38 thf(fact_52_curry__case__prod,axiom,
% 0.17/0.38 ! [C: $tType,B: $tType,A: $tType,F4: A > B > C] :
% 0.17/0.38 ( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F4 ) )
% 0.17/0.38 = F4 ) ).
% 0.17/0.38
% 0.17/0.38 % curry_case_prod
% 0.17/0.38 thf(fact_53_case__prod__curry,axiom,
% 0.17/0.38 ! [C: $tType,B: $tType,A: $tType,F4: ( product_prod @ A @ B ) > C] :
% 0.17/0.38 ( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F4 ) )
% 0.17/0.38 = F4 ) ).
% 0.17/0.38
% 0.17/0.38 % case_prod_curry
% 0.17/0.38 thf(fact_54_mem__case__prodE,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),P2: product_prod @ B @ C] :
% 0.17/0.38 ( ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P2 ) )
% 0.17/0.38 => ~ ! [X3: B,Y3: C] :
% 0.17/0.38 ( ( P2
% 0.17/0.38 = ( product_Pair @ B @ C @ X3 @ Y3 ) )
% 0.17/0.38 => ~ ( member @ A @ Z2 @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % mem_case_prodE
% 0.17/0.38 thf(fact_55_old_Oprod_Ocase,axiom,
% 0.17/0.38 ! [A: $tType,C: $tType,B: $tType,F4: A > B > C,X1: A,X2: B] :
% 0.17/0.38 ( ( product_case_prod @ A @ B @ C @ F4 @ ( product_Pair @ A @ B @ X1 @ X2 ) )
% 0.17/0.38 = ( F4 @ X1 @ X2 ) ) ).
% 0.17/0.38
% 0.17/0.38 % old.prod.case
% 0.17/0.38 thf(fact_56_dtree__cong,axiom,
% 0.17/0.38 ! [Tr: dtree,Tr3: dtree] :
% 0.17/0.38 ( ( ( root @ Tr )
% 0.17/0.38 = ( root @ Tr3 ) )
% 0.17/0.38 => ( ( ( cont @ Tr )
% 0.17/0.38 = ( cont @ Tr3 ) )
% 0.17/0.38 => ( Tr = Tr3 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % dtree_cong
% 0.17/0.38 thf(fact_57_curryE,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,F4: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
% 0.17/0.38 ( ( product_curry @ A @ B @ $o @ F4 @ A3 @ B3 )
% 0.17/0.38 => ( F4 @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % curryE
% 0.17/0.38 thf(fact_58_curryD,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,F4: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
% 0.17/0.38 ( ( product_curry @ A @ B @ $o @ F4 @ A3 @ B3 )
% 0.17/0.38 => ( F4 @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % curryD
% 0.17/0.38 thf(fact_59_finite__code,axiom,
% 0.17/0.38 ! [A: $tType] :
% 0.17/0.38 ( ( finite_finite @ A @ ( type2 @ A ) )
% 0.17/0.38 => ( ( finite_finite2 @ A )
% 0.17/0.38 = ( ^ [A7: set @ A] : $true ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % finite_code
% 0.17/0.38 thf(fact_60_split__cong,axiom,
% 0.17/0.38 ! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F4: A > B > C,G3: A > B > C,P2: product_prod @ A @ B] :
% 0.17/0.38 ( ! [X3: A,Y3: B] :
% 0.17/0.38 ( ( ( product_Pair @ A @ B @ X3 @ Y3 )
% 0.17/0.38 = Q2 )
% 0.17/0.38 => ( ( F4 @ X3 @ Y3 )
% 0.17/0.38 = ( G3 @ X3 @ Y3 ) ) )
% 0.17/0.38 => ( ( P2 = Q2 )
% 0.17/0.38 => ( ( product_case_prod @ A @ B @ C @ F4 @ P2 )
% 0.17/0.38 = ( product_case_prod @ A @ B @ C @ G3 @ Q2 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % split_cong
% 0.17/0.38 thf(fact_61_same__fstI,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,P: A > $o,X: A,Y6: B,Y: B,R3: A > ( set @ ( product_prod @ B @ B ) )] :
% 0.17/0.38 ( ( P @ X )
% 0.17/0.38 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y6 @ Y ) @ ( R3 @ X ) )
% 0.17/0.38 => ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y6 ) @ ( product_Pair @ A @ B @ X @ Y ) ) @ ( same_fst @ A @ B @ P @ R3 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % same_fstI
% 0.17/0.38 thf(fact_62_cont__Node,axiom,
% 0.17/0.38 ! [As: set @ ( sum_sum @ t @ dtree ),N: n] :
% 0.17/0.38 ( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As )
% 0.17/0.38 => ( ( cont @ ( node @ N @ As ) )
% 0.17/0.38 = As ) ) ).
% 0.17/0.38
% 0.17/0.38 % cont_Node
% 0.17/0.38 thf(fact_63_finite__set__choice,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,A6: set @ A,P: A > B > $o] :
% 0.17/0.38 ( ( finite_finite2 @ A @ A6 )
% 0.17/0.38 => ( ! [X3: A] :
% 0.17/0.38 ( ( member @ A @ X3 @ A6 )
% 0.17/0.38 => ? [X12: B] : ( P @ X3 @ X12 ) )
% 0.17/0.38 => ? [F2: A > B] :
% 0.17/0.38 ! [X5: A] :
% 0.17/0.38 ( ( member @ A @ X5 @ A6 )
% 0.17/0.38 => ( P @ X5 @ ( F2 @ X5 ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % finite_set_choice
% 0.17/0.38 thf(fact_64_finite,axiom,
% 0.17/0.38 ! [A: $tType] :
% 0.17/0.38 ( ( finite_finite @ A @ ( type2 @ A ) )
% 0.17/0.38 => ! [A6: set @ A] : ( finite_finite2 @ A @ A6 ) ) ).
% 0.17/0.38
% 0.17/0.38 % finite
% 0.17/0.38 thf(fact_65_scomp__apply,axiom,
% 0.17/0.38 ! [A: $tType,D: $tType,C: $tType,B: $tType] :
% 0.17/0.38 ( ( product_scomp @ B @ C @ D @ A )
% 0.17/0.38 = ( ^ [F3: B > ( product_prod @ C @ D ),G4: C > D > A,X4: B] : ( product_case_prod @ C @ D @ A @ G4 @ ( F3 @ X4 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % scomp_apply
% 0.17/0.38 thf(fact_66_pair__in__swap__image,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,Y: A,X: B,A6: set @ ( product_prod @ B @ A )] :
% 0.17/0.38 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A6 ) )
% 0.17/0.38 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A6 ) ) ).
% 0.17/0.38
% 0.17/0.38 % pair_in_swap_image
% 0.17/0.38 thf(fact_67_finite__imageI,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,F5: set @ A,H: A > B] :
% 0.17/0.38 ( ( finite_finite2 @ A @ F5 )
% 0.17/0.38 => ( finite_finite2 @ B @ ( image @ A @ B @ H @ F5 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % finite_imageI
% 0.17/0.38 thf(fact_68_dtree__simps,axiom,
% 0.17/0.38 ! [As: set @ ( sum_sum @ t @ dtree ),As2: set @ ( sum_sum @ t @ dtree ),N: n,N2: n] :
% 0.17/0.38 ( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As )
% 0.17/0.38 => ( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As2 )
% 0.17/0.38 => ( ( ( node @ N @ As )
% 0.17/0.38 = ( node @ N2 @ As2 ) )
% 0.17/0.38 = ( ( N = N2 )
% 0.17/0.38 & ( As = As2 ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % dtree_simps
% 0.17/0.38 thf(fact_69_finite__cont,axiom,
% 0.17/0.38 ! [Tr: dtree] : ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ ( cont @ Tr ) ) ).
% 0.17/0.38
% 0.17/0.38 % finite_cont
% 0.17/0.38 thf(fact_70_dtree__cases,axiom,
% 0.17/0.38 ! [Tr: dtree] :
% 0.17/0.38 ~ ! [N3: n,As3: set @ ( sum_sum @ t @ dtree )] :
% 0.17/0.38 ( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As3 )
% 0.17/0.38 => ( Tr
% 0.17/0.38 != ( node @ N3 @ As3 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % dtree_cases
% 0.17/0.38 thf(fact_71_scomp__Pair,axiom,
% 0.17/0.38 ! [C: $tType,B: $tType,A: $tType,X: A > ( product_prod @ B @ C )] :
% 0.17/0.38 ( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X @ ( product_Pair @ B @ C ) )
% 0.17/0.38 = X ) ).
% 0.17/0.38
% 0.17/0.38 % scomp_Pair
% 0.17/0.38 thf(fact_72_Pair__scomp,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,C: $tType,X: C,F4: C > A > B] :
% 0.17/0.38 ( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X ) @ F4 )
% 0.17/0.38 = ( F4 @ X ) ) ).
% 0.17/0.38
% 0.17/0.38 % Pair_scomp
% 0.17/0.38 thf(fact_73_scomp__def,axiom,
% 0.17/0.38 ! [D: $tType,C: $tType,B: $tType,A: $tType] :
% 0.17/0.38 ( ( product_scomp @ A @ B @ C @ D )
% 0.17/0.38 = ( ^ [F3: A > ( product_prod @ B @ C ),G4: B > C > D,X4: A] : ( product_case_prod @ B @ C @ D @ G4 @ ( F3 @ X4 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % scomp_def
% 0.17/0.38 thf(fact_74_image__eqI,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,B3: A,F4: B > A,X: B,A6: set @ B] :
% 0.17/0.38 ( ( B3
% 0.17/0.38 = ( F4 @ X ) )
% 0.17/0.38 => ( ( member @ B @ X @ A6 )
% 0.17/0.38 => ( member @ A @ B3 @ ( image @ B @ A @ F4 @ A6 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % image_eqI
% 0.17/0.38 thf(fact_75_surj__swap,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType] :
% 0.17/0.38 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 0.17/0.38 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % surj_swap
% 0.17/0.38 thf(fact_76_Sup_OSUP__cong,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,A6: set @ B,B6: set @ B,C4: B > A,D3: B > A,Sup: ( set @ A ) > A] :
% 0.17/0.38 ( ( A6 = B6 )
% 0.17/0.38 => ( ! [X3: B] :
% 0.17/0.38 ( ( member @ B @ X3 @ B6 )
% 0.17/0.38 => ( ( C4 @ X3 )
% 0.17/0.38 = ( D3 @ X3 ) ) )
% 0.17/0.38 => ( ( Sup @ ( image @ B @ A @ C4 @ A6 ) )
% 0.17/0.38 = ( Sup @ ( image @ B @ A @ D3 @ B6 ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % Sup.SUP_cong
% 0.17/0.38 thf(fact_77_Inf_OINF__cong,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,A6: set @ B,B6: set @ B,C4: B > A,D3: B > A,Inf: ( set @ A ) > A] :
% 0.17/0.38 ( ( A6 = B6 )
% 0.17/0.38 => ( ! [X3: B] :
% 0.17/0.38 ( ( member @ B @ X3 @ B6 )
% 0.17/0.38 => ( ( C4 @ X3 )
% 0.17/0.38 = ( D3 @ X3 ) ) )
% 0.17/0.38 => ( ( Inf @ ( image @ B @ A @ C4 @ A6 ) )
% 0.17/0.38 = ( Inf @ ( image @ B @ A @ D3 @ B6 ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % Inf.INF_cong
% 0.17/0.38 thf(fact_78_rev__image__eqI,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,X: A,A6: set @ A,B3: B,F4: A > B] :
% 0.17/0.38 ( ( member @ A @ X @ A6 )
% 0.17/0.38 => ( ( B3
% 0.17/0.38 = ( F4 @ X ) )
% 0.17/0.38 => ( member @ B @ B3 @ ( image @ A @ B @ F4 @ A6 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % rev_image_eqI
% 0.17/0.38 thf(fact_79_ball__imageD,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B,P: A > $o] :
% 0.17/0.38 ( ! [X3: A] :
% 0.17/0.38 ( ( member @ A @ X3 @ ( image @ B @ A @ F4 @ A6 ) )
% 0.17/0.38 => ( P @ X3 ) )
% 0.17/0.38 => ! [X5: B] :
% 0.17/0.38 ( ( member @ B @ X5 @ A6 )
% 0.17/0.38 => ( P @ ( F4 @ X5 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % ball_imageD
% 0.17/0.38 thf(fact_80_image__cong,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,M: set @ A,N4: set @ A,F4: A > B,G3: A > B] :
% 0.17/0.38 ( ( M = N4 )
% 0.17/0.38 => ( ! [X3: A] :
% 0.17/0.38 ( ( member @ A @ X3 @ N4 )
% 0.17/0.38 => ( ( F4 @ X3 )
% 0.17/0.38 = ( G3 @ X3 ) ) )
% 0.17/0.38 => ( ( image @ A @ B @ F4 @ M )
% 0.17/0.38 = ( image @ A @ B @ G3 @ N4 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % image_cong
% 0.17/0.38 thf(fact_81_UNIV__I,axiom,
% 0.17/0.38 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % UNIV_I
% 0.17/0.38 thf(fact_82_finite__Plus__UNIV__iff,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType] :
% 0.17/0.38 ( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) )
% 0.17/0.38 = ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
% 0.17/0.38 & ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % finite_Plus_UNIV_iff
% 0.17/0.38 thf(fact_83_Finite__Set_Ofinite__set,axiom,
% 0.17/0.38 ! [A: $tType] :
% 0.17/0.38 ( ( finite_finite2 @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
% 0.17/0.38 = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % Finite_Set.finite_set
% 0.17/0.38 thf(fact_84_finite__prod,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType] :
% 0.17/0.38 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.17/0.38 = ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
% 0.17/0.38 & ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % finite_prod
% 0.17/0.38 thf(fact_85_finite__Prod__UNIV,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType] :
% 0.17/0.38 ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
% 0.17/0.38 => ( ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) )
% 0.17/0.38 => ( finite_finite2 @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % finite_Prod_UNIV
% 0.17/0.38 thf(fact_86_finite__fun__UNIVD2,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType] :
% 0.17/0.38 ( ( finite_finite2 @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) )
% 0.17/0.38 => ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % finite_fun_UNIVD2
% 0.17/0.38 thf(fact_87_range__eqI,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,B3: A,F4: B > A,X: B] :
% 0.17/0.38 ( ( B3
% 0.17/0.38 = ( F4 @ X ) )
% 0.17/0.38 => ( member @ A @ B3 @ ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % range_eqI
% 0.17/0.38 thf(fact_88_rangeI,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,F4: B > A,X: B] : ( member @ A @ ( F4 @ X ) @ ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % rangeI
% 0.17/0.38 thf(fact_89_UNIV__eq__I,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A] :
% 0.17/0.38 ( ! [X3: A] : ( member @ A @ X3 @ A6 )
% 0.17/0.38 => ( ( top_top @ ( set @ A ) )
% 0.17/0.38 = A6 ) ) ).
% 0.17/0.38
% 0.17/0.38 % UNIV_eq_I
% 0.17/0.38 thf(fact_90_UNIV__witness,axiom,
% 0.17/0.38 ! [A: $tType] :
% 0.17/0.38 ? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % UNIV_witness
% 0.17/0.38 thf(fact_91_finite__UNIV,axiom,
% 0.17/0.38 ! [A: $tType] :
% 0.17/0.38 ( ( finite_finite @ A @ ( type2 @ A ) )
% 0.17/0.38 => ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % finite_UNIV
% 0.17/0.38 thf(fact_92_infinite__UNIV__char__0,axiom,
% 0.17/0.38 ! [A: $tType] :
% 0.17/0.38 ( ( semiring_char_0 @ A @ ( type2 @ A ) )
% 0.17/0.38 => ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % infinite_UNIV_char_0
% 0.17/0.38 thf(fact_93_ex__new__if__finite,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A] :
% 0.17/0.38 ( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
% 0.17/0.38 => ( ( finite_finite2 @ A @ A6 )
% 0.17/0.38 => ? [A5: A] :
% 0.17/0.38 ~ ( member @ A @ A5 @ A6 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % ex_new_if_finite
% 0.17/0.38 thf(fact_94_imageI,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,X: A,A6: set @ A,F4: A > B] :
% 0.17/0.38 ( ( member @ A @ X @ A6 )
% 0.17/0.38 => ( member @ B @ ( F4 @ X ) @ ( image @ A @ B @ F4 @ A6 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % imageI
% 0.17/0.38 thf(fact_95_image__iff,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,Z2: A,F4: B > A,A6: set @ B] :
% 0.17/0.38 ( ( member @ A @ Z2 @ ( image @ B @ A @ F4 @ A6 ) )
% 0.17/0.38 = ( ? [X4: B] :
% 0.17/0.38 ( ( member @ B @ X4 @ A6 )
% 0.17/0.38 & ( Z2
% 0.17/0.38 = ( F4 @ X4 ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % image_iff
% 0.17/0.38 thf(fact_96_bex__imageD,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B,P: A > $o] :
% 0.17/0.38 ( ? [X5: A] :
% 0.17/0.38 ( ( member @ A @ X5 @ ( image @ B @ A @ F4 @ A6 ) )
% 0.17/0.38 & ( P @ X5 ) )
% 0.17/0.38 => ? [X3: B] :
% 0.17/0.38 ( ( member @ B @ X3 @ A6 )
% 0.17/0.38 & ( P @ ( F4 @ X3 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % bex_imageD
% 0.17/0.38 thf(fact_97_finite__option__UNIV,axiom,
% 0.17/0.38 ! [A: $tType] :
% 0.17/0.38 ( ( finite_finite2 @ ( option @ A ) @ ( top_top @ ( set @ ( option @ A ) ) ) )
% 0.17/0.38 = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % finite_option_UNIV
% 0.17/0.38 thf(fact_98_iso__tuple__UNIV__I,axiom,
% 0.17/0.38 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % iso_tuple_UNIV_I
% 0.17/0.38 thf(fact_99_top__apply,axiom,
% 0.17/0.38 ! [C: $tType,D: $tType] :
% 0.17/0.38 ( ( top @ C @ ( type2 @ C ) )
% 0.17/0.38 => ( ( top_top @ ( D > C ) )
% 0.17/0.38 = ( ^ [X4: D] : ( top_top @ C ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % top_apply
% 0.17/0.38 thf(fact_100_surj__def,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,F4: B > A] :
% 0.17/0.38 ( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 0.17/0.38 = ( top_top @ ( set @ A ) ) )
% 0.17/0.38 = ( ! [Y5: A] :
% 0.17/0.38 ? [X4: B] :
% 0.17/0.38 ( Y5
% 0.17/0.38 = ( F4 @ X4 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % surj_def
% 0.17/0.38 thf(fact_101_surjI,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,G3: B > A,F4: A > B] :
% 0.17/0.38 ( ! [X3: A] :
% 0.17/0.38 ( ( G3 @ ( F4 @ X3 ) )
% 0.17/0.38 = X3 )
% 0.17/0.38 => ( ( image @ B @ A @ G3 @ ( top_top @ ( set @ B ) ) )
% 0.17/0.38 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % surjI
% 0.17/0.38 thf(fact_102_surjE,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,F4: B > A,Y: A] :
% 0.17/0.38 ( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 0.17/0.38 = ( top_top @ ( set @ A ) ) )
% 0.17/0.38 => ~ ! [X3: B] :
% 0.17/0.38 ( Y
% 0.17/0.38 != ( F4 @ X3 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % surjE
% 0.17/0.38 thf(fact_103_top__set__def,axiom,
% 0.17/0.38 ! [A: $tType] :
% 0.17/0.38 ( ( top_top @ ( set @ A ) )
% 0.17/0.38 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % top_set_def
% 0.17/0.38 thf(fact_104_surjD,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,F4: B > A,Y: A] :
% 0.17/0.38 ( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 0.17/0.38 = ( top_top @ ( set @ A ) ) )
% 0.17/0.38 => ? [X3: B] :
% 0.17/0.38 ( Y
% 0.17/0.38 = ( F4 @ X3 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % surjD
% 0.17/0.38 thf(fact_105_map__prod__surj,axiom,
% 0.17/0.38 ! [A: $tType,C: $tType,D: $tType,B: $tType,F4: A > B,G3: C > D] :
% 0.17/0.38 ( ( ( image @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.38 = ( top_top @ ( set @ B ) ) )
% 0.17/0.38 => ( ( ( image @ C @ D @ G3 @ ( top_top @ ( set @ C ) ) )
% 0.17/0.38 = ( top_top @ ( set @ D ) ) )
% 0.17/0.38 => ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F4 @ G3 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
% 0.17/0.38 = ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % map_prod_surj
% 0.17/0.38 thf(fact_106_Pow__UNIV,axiom,
% 0.17/0.38 ! [A: $tType] :
% 0.17/0.38 ( ( pow @ A @ ( top_top @ ( set @ A ) ) )
% 0.17/0.38 = ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % Pow_UNIV
% 0.17/0.38 thf(fact_107_surj__swap__iff,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,A3: B,B3: B,F4: B > A] :
% 0.17/0.38 ( ( ( image @ B @ A @ ( swap @ B @ A @ A3 @ B3 @ F4 ) @ ( top_top @ ( set @ B ) ) )
% 0.17/0.38 = ( top_top @ ( set @ A ) ) )
% 0.17/0.38 = ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 0.17/0.38 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % surj_swap_iff
% 0.17/0.38 thf(fact_108_top__empty__eq,axiom,
% 0.17/0.38 ! [A: $tType] :
% 0.17/0.38 ( ( top_top @ ( A > $o ) )
% 0.17/0.38 = ( ^ [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % top_empty_eq
% 0.17/0.38 thf(fact_109_finite__Pow__iff,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A] :
% 0.17/0.38 ( ( finite_finite2 @ ( set @ A ) @ ( pow @ A @ A6 ) )
% 0.17/0.38 = ( finite_finite2 @ A @ A6 ) ) ).
% 0.17/0.38
% 0.17/0.38 % finite_Pow_iff
% 0.17/0.38 thf(fact_110_top1I,axiom,
% 0.17/0.38 ! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% 0.17/0.38
% 0.17/0.38 % top1I
% 0.17/0.38 thf(fact_111_map__prod__simp,axiom,
% 0.17/0.38 ! [C: $tType,A: $tType,B: $tType,D: $tType,F4: C > A,G3: D > B,A3: C,B3: D] :
% 0.17/0.38 ( ( product_map_prod @ C @ A @ D @ B @ F4 @ G3 @ ( product_Pair @ C @ D @ A3 @ B3 ) )
% 0.17/0.38 = ( product_Pair @ A @ B @ ( F4 @ A3 ) @ ( G3 @ B3 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % map_prod_simp
% 0.17/0.38 thf(fact_112_swap__image__eq,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,A3: A,A6: set @ A,B3: A,F4: A > B] :
% 0.17/0.38 ( ( member @ A @ A3 @ A6 )
% 0.17/0.38 => ( ( member @ A @ B3 @ A6 )
% 0.17/0.38 => ( ( image @ A @ B @ ( swap @ A @ B @ A3 @ B3 @ F4 ) @ A6 )
% 0.17/0.38 = ( image @ A @ B @ F4 @ A6 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % swap_image_eq
% 0.17/0.38 thf(fact_113_map__prod__imageI,axiom,
% 0.17/0.38 ! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B3: B,R3: set @ ( product_prod @ A @ B ),F4: A > C,G3: B > D] :
% 0.17/0.38 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R3 )
% 0.17/0.38 => ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F4 @ A3 ) @ ( G3 @ B3 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F4 @ G3 ) @ R3 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % map_prod_imageI
% 0.17/0.38 thf(fact_114_Cantors__paradox,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A] :
% 0.17/0.38 ~ ? [X5: A > ( set @ A )] :
% 0.17/0.38 ( ( image @ A @ ( set @ A ) @ X5 @ A6 )
% 0.17/0.38 = ( pow @ A @ A6 ) ) ).
% 0.17/0.38
% 0.17/0.38 % Cantors_paradox
% 0.17/0.38 thf(fact_115_Pow__top,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A] : ( member @ ( set @ A ) @ A6 @ ( pow @ A @ A6 ) ) ).
% 0.17/0.38
% 0.17/0.38 % Pow_top
% 0.17/0.38 thf(fact_116_image__Pow__surj,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,F4: B > A,A6: set @ B,B6: set @ A] :
% 0.17/0.38 ( ( ( image @ B @ A @ F4 @ A6 )
% 0.17/0.38 = B6 )
% 0.17/0.38 => ( ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F4 ) @ ( pow @ B @ A6 ) )
% 0.17/0.38 = ( pow @ A @ B6 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % image_Pow_surj
% 0.17/0.38 thf(fact_117_prod__fun__imageE,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,D: $tType,C: $tType,C2: product_prod @ A @ B,F4: C > A,G3: D > B,R3: set @ ( product_prod @ C @ D )] :
% 0.17/0.38 ( ( member @ ( product_prod @ A @ B ) @ C2 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F4 @ G3 ) @ R3 ) )
% 0.17/0.38 => ~ ! [X3: C,Y3: D] :
% 0.17/0.38 ( ( C2
% 0.17/0.38 = ( product_Pair @ A @ B @ ( F4 @ X3 ) @ ( G3 @ Y3 ) ) )
% 0.17/0.38 => ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X3 @ Y3 ) @ R3 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % prod_fun_imageE
% 0.17/0.38 thf(fact_118_surj__imp__surj__swap,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,F4: B > A,A3: B,B3: B] :
% 0.17/0.38 ( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 0.17/0.38 = ( top_top @ ( set @ A ) ) )
% 0.17/0.38 => ( ( image @ B @ A @ ( swap @ B @ A @ A3 @ B3 @ F4 ) @ ( top_top @ ( set @ B ) ) )
% 0.17/0.38 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % surj_imp_surj_swap
% 0.17/0.38 thf(fact_119_top__conj_I2_J,axiom,
% 0.17/0.38 ! [A: $tType,P: $o,X: A] :
% 0.17/0.38 ( ( P
% 0.17/0.38 & ( top_top @ ( A > $o ) @ X ) )
% 0.17/0.38 = P ) ).
% 0.17/0.38
% 0.17/0.38 % top_conj(2)
% 0.17/0.38 thf(fact_120_top__conj_I1_J,axiom,
% 0.17/0.38 ! [A: $tType,X: A,P: $o] :
% 0.17/0.38 ( ( ( top_top @ ( A > $o ) @ X )
% 0.17/0.38 & P )
% 0.17/0.38 = P ) ).
% 0.17/0.38
% 0.17/0.38 % top_conj(1)
% 0.17/0.38 thf(fact_121_range__fst,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType] :
% 0.17/0.38 ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.17/0.38 = ( top_top @ ( set @ A ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % range_fst
% 0.17/0.38 thf(fact_122_range__snd,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType] :
% 0.17/0.38 ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 0.17/0.38 = ( top_top @ ( set @ A ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % range_snd
% 0.17/0.38 thf(fact_123_in__inv__image,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,X: A,Y: A,R: set @ ( product_prod @ B @ B ),F4: A > B] :
% 0.17/0.38 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( inv_image @ B @ A @ R @ F4 ) )
% 0.17/0.38 = ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F4 @ X ) @ ( F4 @ Y ) ) @ R ) ) ).
% 0.17/0.38
% 0.17/0.38 % in_inv_image
% 0.17/0.38 thf(fact_124_fst__map__prod,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,D: $tType,C: $tType,F4: C > A,G3: D > B,X: product_prod @ C @ D] :
% 0.17/0.38 ( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F4 @ G3 @ X ) )
% 0.17/0.38 = ( F4 @ ( product_fst @ C @ D @ X ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % fst_map_prod
% 0.17/0.38 thf(fact_125_snd__map__prod,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,D: $tType,C: $tType,F4: C > B,G3: D > A,X: product_prod @ C @ D] :
% 0.17/0.38 ( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F4 @ G3 @ X ) )
% 0.17/0.38 = ( G3 @ ( product_snd @ C @ D @ X ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % snd_map_prod
% 0.17/0.38 thf(fact_126_prod_Ocollapse,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.17/0.38 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 0.17/0.38 = Prod ) ).
% 0.17/0.38
% 0.17/0.38 % prod.collapse
% 0.17/0.38 thf(fact_127_snd__swap,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,X: product_prod @ A @ B] :
% 0.17/0.38 ( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
% 0.17/0.38 = ( product_fst @ A @ B @ X ) ) ).
% 0.17/0.38
% 0.17/0.38 % snd_swap
% 0.17/0.38 thf(fact_128_fst__swap,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,X: product_prod @ B @ A] :
% 0.17/0.38 ( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
% 0.17/0.38 = ( product_snd @ B @ A @ X ) ) ).
% 0.17/0.38
% 0.17/0.38 % fst_swap
% 0.17/0.38 thf(fact_129_prod_Osplit__sel,axiom,
% 0.17/0.38 ! [C: $tType,B: $tType,A: $tType,P: C > $o,F4: A > B > C,Prod: product_prod @ A @ B] :
% 0.17/0.38 ( ( P @ ( product_case_prod @ A @ B @ C @ F4 @ Prod ) )
% 0.17/0.38 = ( ( Prod
% 0.17/0.38 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 0.17/0.38 => ( P @ ( F4 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % prod.split_sel
% 0.17/0.38 thf(fact_130_prod_Osplit__sel__asm,axiom,
% 0.17/0.38 ! [C: $tType,B: $tType,A: $tType,P: C > $o,F4: A > B > C,Prod: product_prod @ A @ B] :
% 0.17/0.38 ( ( P @ ( product_case_prod @ A @ B @ C @ F4 @ Prod ) )
% 0.17/0.38 = ( ~ ( ( Prod
% 0.17/0.38 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 0.17/0.38 & ~ ( P @ ( F4 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % prod.split_sel_asm
% 0.17/0.38 thf(fact_131_prod_Oexhaust__sel,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.17/0.38 ( Prod
% 0.17/0.38 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % prod.exhaust_sel
% 0.17/0.38 thf(fact_132_surjective__pairing,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 0.17/0.38 ( T2
% 0.17/0.38 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % surjective_pairing
% 0.17/0.38 thf(fact_133_prod__eqI,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q2: product_prod @ A @ B] :
% 0.17/0.38 ( ( ( product_fst @ A @ B @ P2 )
% 0.17/0.38 = ( product_fst @ A @ B @ Q2 ) )
% 0.17/0.38 => ( ( ( product_snd @ A @ B @ P2 )
% 0.17/0.38 = ( product_snd @ A @ B @ Q2 ) )
% 0.17/0.38 => ( P2 = Q2 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % prod_eqI
% 0.17/0.38 thf(fact_134_prod_Oexpand,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
% 0.17/0.38 ( ( ( ( product_fst @ A @ B @ Prod )
% 0.17/0.38 = ( product_fst @ A @ B @ Prod2 ) )
% 0.17/0.38 & ( ( product_snd @ A @ B @ Prod )
% 0.17/0.38 = ( product_snd @ A @ B @ Prod2 ) ) )
% 0.17/0.38 => ( Prod = Prod2 ) ) ).
% 0.17/0.38
% 0.17/0.38 % prod.expand
% 0.17/0.38 thf(fact_135_prod__eq__iff,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType] :
% 0.17/0.38 ( ( ^ [Y7: product_prod @ A @ B,Z3: product_prod @ A @ B] : ( Y7 = Z3 ) )
% 0.17/0.38 = ( ^ [S3: product_prod @ A @ B,T3: product_prod @ A @ B] :
% 0.17/0.38 ( ( ( product_fst @ A @ B @ S3 )
% 0.17/0.38 = ( product_fst @ A @ B @ T3 ) )
% 0.17/0.38 & ( ( product_snd @ A @ B @ S3 )
% 0.17/0.38 = ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % prod_eq_iff
% 0.17/0.38 thf(fact_136_case__prod__beta,axiom,
% 0.17/0.38 ! [A: $tType,C: $tType,B: $tType] :
% 0.17/0.38 ( ( product_case_prod @ B @ C @ A )
% 0.17/0.38 = ( ^ [F3: B > C > A,P3: product_prod @ B @ C] : ( F3 @ ( product_fst @ B @ C @ P3 ) @ ( product_snd @ B @ C @ P3 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % case_prod_beta
% 0.17/0.38 thf(fact_137_prod_Ocase__eq__if,axiom,
% 0.17/0.38 ! [C: $tType,B: $tType,A: $tType] :
% 0.17/0.38 ( ( product_case_prod @ A @ B @ C )
% 0.17/0.38 = ( ^ [F3: A > B > C,Prod3: product_prod @ A @ B] : ( F3 @ ( product_fst @ A @ B @ Prod3 ) @ ( product_snd @ A @ B @ Prod3 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % prod.case_eq_if
% 0.17/0.38 thf(fact_138_Product__Type_OCollect__case__prodD,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,X: product_prod @ A @ B,A6: A > B > $o] :
% 0.17/0.38 ( ( member @ ( product_prod @ A @ B ) @ X @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A6 ) ) )
% 0.17/0.38 => ( A6 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % Product_Type.Collect_case_prodD
% 0.17/0.38 thf(fact_139_fst__conv,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,X1: A,X2: B] :
% 0.17/0.38 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X2 ) )
% 0.17/0.38 = X1 ) ).
% 0.17/0.38
% 0.17/0.38 % fst_conv
% 0.17/0.38 thf(fact_140_fst__eqD,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,X: A,Y: B,A3: A] :
% 0.17/0.38 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
% 0.17/0.38 = A3 )
% 0.17/0.38 => ( X = A3 ) ) ).
% 0.17/0.38
% 0.17/0.38 % fst_eqD
% 0.17/0.38 thf(fact_141_snd__conv,axiom,
% 0.17/0.38 ! [Aa: $tType,A: $tType,X1: Aa,X2: A] :
% 0.17/0.38 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X2 ) )
% 0.17/0.38 = X2 ) ).
% 0.17/0.38
% 0.17/0.38 % snd_conv
% 0.17/0.38 thf(fact_142_snd__eqD,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,X: B,Y: A,A3: A] :
% 0.17/0.38 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 0.17/0.38 = A3 )
% 0.17/0.38 => ( Y = A3 ) ) ).
% 0.17/0.38
% 0.17/0.38 % snd_eqD
% 0.17/0.38 thf(fact_143_prod_Oswap__def,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType] :
% 0.17/0.38 ( ( product_swap @ A @ B )
% 0.17/0.38 = ( ^ [P3: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P3 ) @ ( product_fst @ A @ B @ P3 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % prod.swap_def
% 0.17/0.38 thf(fact_144_exI__realizer,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,P: A > B > $o,Y: A,X: B] :
% 0.17/0.38 ( ( P @ Y @ X )
% 0.17/0.38 => ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % exI_realizer
% 0.17/0.38 thf(fact_145_conjI__realizer,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,P: A > $o,P2: A,Q: B > $o,Q2: B] :
% 0.17/0.38 ( ( P @ P2 )
% 0.17/0.38 => ( ( Q @ Q2 )
% 0.17/0.38 => ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) )
% 0.17/0.38 & ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % conjI_realizer
% 0.17/0.38 thf(fact_146_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,P: A > B > $o,X: A,Y: B,A3: product_prod @ A @ B] :
% 0.17/0.38 ( ( P @ X @ Y )
% 0.17/0.38 => ( ( A3
% 0.17/0.38 = ( product_Pair @ A @ B @ X @ Y ) )
% 0.17/0.38 => ( P @ ( product_fst @ A @ B @ A3 ) @ ( product_snd @ A @ B @ A3 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % BNF_Greatest_Fixpoint.subst_Pair
% 0.17/0.38 thf(fact_147_sndI,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,X: product_prod @ A @ B,Y: A,Z2: B] :
% 0.17/0.38 ( ( X
% 0.17/0.38 = ( product_Pair @ A @ B @ Y @ Z2 ) )
% 0.17/0.38 => ( ( product_snd @ A @ B @ X )
% 0.17/0.38 = Z2 ) ) ).
% 0.17/0.38
% 0.17/0.38 % sndI
% 0.17/0.38 thf(fact_148_fstI,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,X: product_prod @ A @ B,Y: A,Z2: B] :
% 0.17/0.38 ( ( X
% 0.17/0.38 = ( product_Pair @ A @ B @ Y @ Z2 ) )
% 0.17/0.38 => ( ( product_fst @ A @ B @ X )
% 0.17/0.38 = Y ) ) ).
% 0.17/0.38
% 0.17/0.38 % fstI
% 0.17/0.38 thf(fact_149_eq__snd__iff,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,B3: A,P2: product_prod @ B @ A] :
% 0.17/0.38 ( ( B3
% 0.17/0.38 = ( product_snd @ B @ A @ P2 ) )
% 0.17/0.38 = ( ? [A2: B] :
% 0.17/0.38 ( P2
% 0.17/0.38 = ( product_Pair @ B @ A @ A2 @ B3 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % eq_snd_iff
% 0.17/0.38 thf(fact_150_eq__fst__iff,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,A3: A,P2: product_prod @ A @ B] :
% 0.17/0.38 ( ( A3
% 0.17/0.38 = ( product_fst @ A @ B @ P2 ) )
% 0.17/0.38 = ( ? [B2: B] :
% 0.17/0.38 ( P2
% 0.17/0.38 = ( product_Pair @ A @ B @ A3 @ B2 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % eq_fst_iff
% 0.17/0.38 thf(fact_151_Collect__split__mono__strong,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,X6: set @ A,A6: set @ ( product_prod @ A @ B ),Y8: set @ B,P: A > B > $o,Q: A > B > $o] :
% 0.17/0.38 ( ( X6
% 0.17/0.38 = ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A6 ) )
% 0.17/0.38 => ( ( Y8
% 0.17/0.38 = ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A6 ) )
% 0.17/0.38 => ( ! [X3: A] :
% 0.17/0.38 ( ( member @ A @ X3 @ X6 )
% 0.17/0.38 => ! [Xa: B] :
% 0.17/0.38 ( ( member @ B @ Xa @ Y8 )
% 0.17/0.38 => ( ( P @ X3 @ Xa )
% 0.17/0.38 => ( Q @ X3 @ Xa ) ) ) )
% 0.17/0.38 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) )
% 0.17/0.38 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q ) ) ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % Collect_split_mono_strong
% 0.17/0.38 thf(fact_152_subsetI,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 0.17/0.38 ( ! [X3: A] :
% 0.17/0.38 ( ( member @ A @ X3 @ A6 )
% 0.17/0.38 => ( member @ A @ X3 @ B6 ) )
% 0.17/0.38 => ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) ) ).
% 0.17/0.38
% 0.17/0.38 % subsetI
% 0.17/0.38 thf(fact_153_subset__antisym,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 0.17/0.38 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.17/0.38 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A6 )
% 0.17/0.38 => ( A6 = B6 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % subset_antisym
% 0.17/0.38 thf(fact_154_Pow__iff,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 0.17/0.38 ( ( member @ ( set @ A ) @ A6 @ ( pow @ A @ B6 ) )
% 0.17/0.38 = ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) ) ).
% 0.17/0.38
% 0.17/0.38 % Pow_iff
% 0.17/0.38 thf(fact_155_PowI,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 0.17/0.38 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.17/0.38 => ( member @ ( set @ A ) @ A6 @ ( pow @ A @ B6 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % PowI
% 0.17/0.38 thf(fact_156_subrelI,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 0.17/0.38 ( ! [X3: A,Y3: B] :
% 0.17/0.38 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R )
% 0.17/0.38 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S ) )
% 0.17/0.38 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% 0.17/0.38
% 0.17/0.38 % subrelI
% 0.17/0.38 thf(fact_157_Pow__mono,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 0.17/0.38 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.17/0.38 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow @ A @ A6 ) @ ( pow @ A @ B6 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % Pow_mono
% 0.17/0.38 thf(fact_158_top_Oextremum__uniqueI,axiom,
% 0.17/0.38 ! [A: $tType] :
% 0.17/0.38 ( ( order_top @ A @ ( type2 @ A ) )
% 0.17/0.38 => ! [A3: A] :
% 0.17/0.38 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 0.17/0.38 => ( A3
% 0.17/0.38 = ( top_top @ A ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % top.extremum_uniqueI
% 0.17/0.38 thf(fact_159_top_Oextremum__unique,axiom,
% 0.17/0.38 ! [A: $tType] :
% 0.17/0.38 ( ( order_top @ A @ ( type2 @ A ) )
% 0.17/0.38 => ! [A3: A] :
% 0.17/0.38 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 0.17/0.38 = ( A3
% 0.17/0.38 = ( top_top @ A ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % top.extremum_unique
% 0.17/0.38 thf(fact_160_top__greatest,axiom,
% 0.17/0.38 ! [A: $tType] :
% 0.17/0.38 ( ( order_top @ A @ ( type2 @ A ) )
% 0.17/0.38 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % top_greatest
% 0.17/0.38 thf(fact_161_rev__finite__subset,axiom,
% 0.17/0.38 ! [A: $tType,B6: set @ A,A6: set @ A] :
% 0.17/0.38 ( ( finite_finite2 @ A @ B6 )
% 0.17/0.38 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.17/0.38 => ( finite_finite2 @ A @ A6 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % rev_finite_subset
% 0.17/0.38 thf(fact_162_infinite__super,axiom,
% 0.17/0.38 ! [A: $tType,S4: set @ A,T4: set @ A] :
% 0.17/0.38 ( ( ord_less_eq @ ( set @ A ) @ S4 @ T4 )
% 0.17/0.38 => ( ~ ( finite_finite2 @ A @ S4 )
% 0.17/0.38 => ~ ( finite_finite2 @ A @ T4 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % infinite_super
% 0.17/0.38 thf(fact_163_finite__subset,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 0.17/0.38 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.17/0.38 => ( ( finite_finite2 @ A @ B6 )
% 0.17/0.38 => ( finite_finite2 @ A @ A6 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % finite_subset
% 0.17/0.38 thf(fact_164_subset__UNIV,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ A6 @ ( top_top @ ( set @ A ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % subset_UNIV
% 0.17/0.38 thf(fact_165_set__mp,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A,B6: set @ A,X: A] :
% 0.17/0.38 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.17/0.38 => ( ( member @ A @ X @ A6 )
% 0.17/0.38 => ( member @ A @ X @ B6 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % set_mp
% 0.17/0.38 thf(fact_166_in__mono,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A,B6: set @ A,X: A] :
% 0.17/0.38 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.17/0.38 => ( ( member @ A @ X @ A6 )
% 0.17/0.38 => ( member @ A @ X @ B6 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % in_mono
% 0.17/0.38 thf(fact_167_subsetD,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A,B6: set @ A,C2: A] :
% 0.17/0.38 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.17/0.38 => ( ( member @ A @ C2 @ A6 )
% 0.17/0.38 => ( member @ A @ C2 @ B6 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % subsetD
% 0.17/0.38 thf(fact_168_subsetCE,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A,B6: set @ A,C2: A] :
% 0.17/0.38 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.17/0.38 => ( ( member @ A @ C2 @ A6 )
% 0.17/0.38 => ( member @ A @ C2 @ B6 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % subsetCE
% 0.17/0.38 thf(fact_169_equalityE,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 0.17/0.38 ( ( A6 = B6 )
% 0.17/0.38 => ~ ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.17/0.38 => ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % equalityE
% 0.17/0.38 thf(fact_170_subset__eq,axiom,
% 0.17/0.38 ! [A: $tType] :
% 0.17/0.38 ( ( ord_less_eq @ ( set @ A ) )
% 0.17/0.38 = ( ^ [A7: set @ A,B7: set @ A] :
% 0.17/0.38 ! [X4: A] :
% 0.17/0.38 ( ( member @ A @ X4 @ A7 )
% 0.17/0.38 => ( member @ A @ X4 @ B7 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % subset_eq
% 0.17/0.38 thf(fact_171_equalityD1,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 0.17/0.38 ( ( A6 = B6 )
% 0.17/0.38 => ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) ) ).
% 0.17/0.38
% 0.17/0.38 % equalityD1
% 0.17/0.38 thf(fact_172_equalityD2,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 0.17/0.38 ( ( A6 = B6 )
% 0.17/0.38 => ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ).
% 0.17/0.38
% 0.17/0.38 % equalityD2
% 0.17/0.38 thf(fact_173_set__rev__mp,axiom,
% 0.17/0.38 ! [A: $tType,X: A,A6: set @ A,B6: set @ A] :
% 0.17/0.38 ( ( member @ A @ X @ A6 )
% 0.17/0.38 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.17/0.38 => ( member @ A @ X @ B6 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % set_rev_mp
% 0.17/0.38 thf(fact_174_subset__iff,axiom,
% 0.17/0.38 ! [A: $tType] :
% 0.17/0.38 ( ( ord_less_eq @ ( set @ A ) )
% 0.17/0.38 = ( ^ [A7: set @ A,B7: set @ A] :
% 0.17/0.38 ! [T3: A] :
% 0.17/0.38 ( ( member @ A @ T3 @ A7 )
% 0.17/0.38 => ( member @ A @ T3 @ B7 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % subset_iff
% 0.17/0.38 thf(fact_175_rev__subsetD,axiom,
% 0.17/0.38 ! [A: $tType,C2: A,A6: set @ A,B6: set @ A] :
% 0.17/0.38 ( ( member @ A @ C2 @ A6 )
% 0.17/0.38 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.17/0.38 => ( member @ A @ C2 @ B6 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % rev_subsetD
% 0.17/0.38 thf(fact_176_subset__refl,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ A6 @ A6 ) ).
% 0.17/0.38
% 0.17/0.38 % subset_refl
% 0.17/0.38 thf(fact_177_Collect__mono,axiom,
% 0.17/0.38 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.17/0.38 ( ! [X3: A] :
% 0.17/0.38 ( ( P @ X3 )
% 0.17/0.38 => ( Q @ X3 ) )
% 0.17/0.38 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % Collect_mono
% 0.17/0.38 thf(fact_178_subset__trans,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A,B6: set @ A,C4: set @ A] :
% 0.17/0.38 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.17/0.38 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
% 0.17/0.38 => ( ord_less_eq @ ( set @ A ) @ A6 @ C4 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % subset_trans
% 0.17/0.38 thf(fact_179_set__eq__subset,axiom,
% 0.17/0.38 ! [A: $tType] :
% 0.17/0.38 ( ( ^ [Y7: set @ A,Z3: set @ A] : ( Y7 = Z3 ) )
% 0.17/0.38 = ( ^ [A7: set @ A,B7: set @ A] :
% 0.17/0.38 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 0.17/0.38 & ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % set_eq_subset
% 0.17/0.38 thf(fact_180_contra__subsetD,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A,B6: set @ A,C2: A] :
% 0.17/0.38 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.17/0.38 => ( ~ ( member @ A @ C2 @ B6 )
% 0.17/0.38 => ~ ( member @ A @ C2 @ A6 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % contra_subsetD
% 0.17/0.38 thf(fact_181_Collect__mono__iff,axiom,
% 0.17/0.38 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.17/0.38 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.17/0.38 = ( ! [X4: A] :
% 0.17/0.38 ( ( P @ X4 )
% 0.17/0.38 => ( Q @ X4 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % Collect_mono_iff
% 0.17/0.38 thf(fact_182_PowD,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 0.17/0.38 ( ( member @ ( set @ A ) @ A6 @ ( pow @ A @ B6 ) )
% 0.17/0.38 => ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) ) ).
% 0.17/0.38
% 0.17/0.38 % PowD
% 0.17/0.38 thf(fact_183_image__mono,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,A6: set @ A,B6: set @ A,F4: A > B] :
% 0.17/0.38 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.17/0.38 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) @ ( image @ A @ B @ F4 @ B6 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % image_mono
% 0.17/0.38 thf(fact_184_image__subsetI,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,A6: set @ A,F4: A > B,B6: set @ B] :
% 0.17/0.38 ( ! [X3: A] :
% 0.17/0.38 ( ( member @ A @ X3 @ A6 )
% 0.17/0.38 => ( member @ B @ ( F4 @ X3 ) @ B6 ) )
% 0.17/0.38 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) @ B6 ) ) ).
% 0.17/0.38
% 0.17/0.38 % image_subsetI
% 0.17/0.38 thf(fact_185_subset__imageE,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,B6: set @ A,F4: B > A,A6: set @ B] :
% 0.17/0.38 ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F4 @ A6 ) )
% 0.17/0.38 => ~ ! [C5: set @ B] :
% 0.17/0.38 ( ( ord_less_eq @ ( set @ B ) @ C5 @ A6 )
% 0.17/0.38 => ( B6
% 0.17/0.38 != ( image @ B @ A @ F4 @ C5 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % subset_imageE
% 0.17/0.38 thf(fact_186_image__subset__iff,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B,B6: set @ A] :
% 0.17/0.38 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F4 @ A6 ) @ B6 )
% 0.17/0.38 = ( ! [X4: B] :
% 0.17/0.38 ( ( member @ B @ X4 @ A6 )
% 0.17/0.38 => ( member @ A @ ( F4 @ X4 ) @ B6 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % image_subset_iff
% 0.17/0.38 thf(fact_187_subset__image__iff,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,B6: set @ A,F4: B > A,A6: set @ B] :
% 0.17/0.38 ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F4 @ A6 ) )
% 0.17/0.38 = ( ? [AA: set @ B] :
% 0.17/0.38 ( ( ord_less_eq @ ( set @ B ) @ AA @ A6 )
% 0.17/0.38 & ( B6
% 0.17/0.38 = ( image @ B @ A @ F4 @ AA ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % subset_image_iff
% 0.17/0.38 thf(fact_188_image__Pow__mono,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,F4: B > A,A6: set @ B,B6: set @ A] :
% 0.17/0.38 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F4 @ A6 ) @ B6 )
% 0.17/0.38 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F4 ) @ ( pow @ B @ A6 ) ) @ ( pow @ A @ B6 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % image_Pow_mono
% 0.17/0.38 thf(fact_189_finite__surj,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,A6: set @ A,B6: set @ B,F4: A > B] :
% 0.17/0.38 ( ( finite_finite2 @ A @ A6 )
% 0.17/0.38 => ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( image @ A @ B @ F4 @ A6 ) )
% 0.17/0.38 => ( finite_finite2 @ B @ B6 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % finite_surj
% 0.17/0.38 thf(fact_190_finite__subset__image,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,B6: set @ A,F4: B > A,A6: set @ B] :
% 0.17/0.38 ( ( finite_finite2 @ A @ B6 )
% 0.17/0.38 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F4 @ A6 ) )
% 0.17/0.38 => ? [C5: set @ B] :
% 0.17/0.38 ( ( ord_less_eq @ ( set @ B ) @ C5 @ A6 )
% 0.17/0.38 & ( finite_finite2 @ B @ C5 )
% 0.17/0.38 & ( B6
% 0.17/0.38 = ( image @ B @ A @ F4 @ C5 ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % finite_subset_image
% 0.17/0.38 thf(fact_191_relChain__def,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType] :
% 0.17/0.38 ( ( ord @ B @ ( type2 @ B ) )
% 0.17/0.38 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 0.17/0.38 = ( ^ [R2: set @ ( product_prod @ A @ A ),As4: A > B] :
% 0.17/0.38 ! [I: A,J: A] :
% 0.17/0.38 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R2 )
% 0.17/0.38 => ( ord_less_eq @ B @ ( As4 @ I ) @ ( As4 @ J ) ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % relChain_def
% 0.17/0.38 thf(fact_192_fstOp__def,axiom,
% 0.17/0.38 ! [C: $tType,B: $tType,A: $tType] :
% 0.17/0.38 ( ( bNF_fstOp @ A @ B @ C )
% 0.17/0.38 = ( ^ [P4: A > B > $o,Q3: B > C > $o,Ac: product_prod @ A @ C] : ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Ac ) @ ( bNF_pick_middlep @ A @ B @ C @ P4 @ Q3 @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % fstOp_def
% 0.17/0.38 thf(fact_193_sndOp__def,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,C: $tType] :
% 0.17/0.38 ( ( bNF_sndOp @ C @ A @ B )
% 0.17/0.38 = ( ^ [P4: C > A > $o,Q3: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P4 @ Q3 @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % sndOp_def
% 0.17/0.38 thf(fact_194_accp__subset,axiom,
% 0.17/0.38 ! [A: $tType,R1: A > A > $o,R22: A > A > $o] :
% 0.17/0.38 ( ( ord_less_eq @ ( A > A > $o ) @ R1 @ R22 )
% 0.17/0.38 => ( ord_less_eq @ ( A > $o ) @ ( accp @ A @ R22 ) @ ( accp @ A @ R1 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % accp_subset
% 0.17/0.38 thf(fact_195_accp__subset__induct,axiom,
% 0.17/0.38 ! [A: $tType,D3: A > $o,R3: A > A > $o,X: A,P: A > $o] :
% 0.17/0.38 ( ( ord_less_eq @ ( A > $o ) @ D3 @ ( accp @ A @ R3 ) )
% 0.17/0.38 => ( ! [X3: A,Z: A] :
% 0.17/0.38 ( ( D3 @ X3 )
% 0.17/0.38 => ( ( R3 @ Z @ X3 )
% 0.17/0.38 => ( D3 @ Z ) ) )
% 0.17/0.38 => ( ( D3 @ X )
% 0.17/0.38 => ( ! [X3: A] :
% 0.17/0.38 ( ( D3 @ X3 )
% 0.17/0.38 => ( ! [Z4: A] :
% 0.17/0.38 ( ( R3 @ Z4 @ X3 )
% 0.17/0.38 => ( P @ Z4 ) )
% 0.17/0.38 => ( P @ X3 ) ) )
% 0.17/0.38 => ( P @ X ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % accp_subset_induct
% 0.17/0.38 thf(fact_196_Collect__case__prod__mono,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,A6: A > B > $o,B6: A > B > $o] :
% 0.17/0.38 ( ( ord_less_eq @ ( A > B > $o ) @ A6 @ B6 )
% 0.17/0.38 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A6 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B6 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % Collect_case_prod_mono
% 0.17/0.38 thf(fact_197_finite__vimageD_H,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,F4: A > B,A6: set @ B] :
% 0.17/0.38 ( ( finite_finite2 @ A @ ( vimage @ A @ B @ F4 @ A6 ) )
% 0.17/0.38 => ( ( ord_less_eq @ ( set @ B ) @ A6 @ ( image @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) ) )
% 0.17/0.38 => ( finite_finite2 @ B @ A6 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % finite_vimageD'
% 0.17/0.38 thf(fact_198_image2__eqI,axiom,
% 0.17/0.38 ! [A: $tType,C: $tType,B: $tType,B3: A,F4: B > A,X: B,C2: C,G3: B > C,A6: set @ B] :
% 0.17/0.38 ( ( B3
% 0.17/0.38 = ( F4 @ X ) )
% 0.17/0.38 => ( ( C2
% 0.17/0.38 = ( G3 @ X ) )
% 0.17/0.38 => ( ( member @ B @ X @ A6 )
% 0.17/0.38 => ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B3 @ C2 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A6 @ F4 @ G3 ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % image2_eqI
% 0.17/0.38 thf(fact_199_vimageI,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,F4: B > A,A3: B,B3: A,B6: set @ A] :
% 0.17/0.38 ( ( ( F4 @ A3 )
% 0.17/0.38 = B3 )
% 0.17/0.38 => ( ( member @ A @ B3 @ B6 )
% 0.17/0.38 => ( member @ B @ A3 @ ( vimage @ B @ A @ F4 @ B6 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % vimageI
% 0.17/0.38 thf(fact_200_vimage__eq,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,A3: A,F4: A > B,B6: set @ B] :
% 0.17/0.38 ( ( member @ A @ A3 @ ( vimage @ A @ B @ F4 @ B6 ) )
% 0.17/0.38 = ( member @ B @ ( F4 @ A3 ) @ B6 ) ) ).
% 0.17/0.38
% 0.17/0.38 % vimage_eq
% 0.17/0.38 thf(fact_201_vimage__UNIV,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,F4: A > B] :
% 0.17/0.38 ( ( vimage @ A @ B @ F4 @ ( top_top @ ( set @ B ) ) )
% 0.17/0.38 = ( top_top @ ( set @ A ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % vimage_UNIV
% 0.17/0.38 thf(fact_202_vimage__mono,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,A6: set @ A,B6: set @ A,F4: B > A] :
% 0.17/0.38 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.17/0.38 => ( ord_less_eq @ ( set @ B ) @ ( vimage @ B @ A @ F4 @ A6 ) @ ( vimage @ B @ A @ F4 @ B6 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % vimage_mono
% 0.17/0.38 thf(fact_203_image__subset__iff__subset__vimage,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,F4: B > A,A6: set @ B,B6: set @ A] :
% 0.17/0.38 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F4 @ A6 ) @ B6 )
% 0.17/0.38 = ( ord_less_eq @ ( set @ B ) @ A6 @ ( vimage @ B @ A @ F4 @ B6 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % image_subset_iff_subset_vimage
% 0.17/0.38 thf(fact_204_image__vimage__subset,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,F4: B > A,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F4 @ ( vimage @ B @ A @ F4 @ A6 ) ) @ A6 ) ).
% 0.17/0.38
% 0.17/0.38 % image_vimage_subset
% 0.17/0.38 thf(fact_205_surj__image__vimage__eq,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,F4: B > A,A6: set @ A] :
% 0.17/0.38 ( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 0.17/0.38 = ( top_top @ ( set @ A ) ) )
% 0.17/0.38 => ( ( image @ B @ A @ F4 @ ( vimage @ B @ A @ F4 @ A6 ) )
% 0.17/0.38 = A6 ) ) ).
% 0.17/0.38
% 0.17/0.38 % surj_image_vimage_eq
% 0.17/0.38 thf(fact_206_vimageD,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,A3: A,F4: A > B,A6: set @ B] :
% 0.17/0.38 ( ( member @ A @ A3 @ ( vimage @ A @ B @ F4 @ A6 ) )
% 0.17/0.38 => ( member @ B @ ( F4 @ A3 ) @ A6 ) ) ).
% 0.17/0.38
% 0.17/0.38 % vimageD
% 0.17/0.38 thf(fact_207_vimageE,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,A3: A,F4: A > B,B6: set @ B] :
% 0.17/0.38 ( ( member @ A @ A3 @ ( vimage @ A @ B @ F4 @ B6 ) )
% 0.17/0.38 => ( member @ B @ ( F4 @ A3 ) @ B6 ) ) ).
% 0.17/0.38
% 0.17/0.38 % vimageE
% 0.17/0.38 thf(fact_208_vimageI2,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,F4: B > A,A3: B,A6: set @ A] :
% 0.17/0.38 ( ( member @ A @ ( F4 @ A3 ) @ A6 )
% 0.17/0.38 => ( member @ B @ A3 @ ( vimage @ B @ A @ F4 @ A6 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % vimageI2
% 0.17/0.38 thf(fact_209_vimage__Collect,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,P: B > $o,F4: A > B,Q: A > $o] :
% 0.17/0.38 ( ! [X3: A] :
% 0.17/0.38 ( ( P @ ( F4 @ X3 ) )
% 0.17/0.38 = ( Q @ X3 ) )
% 0.17/0.38 => ( ( vimage @ A @ B @ F4 @ ( collect @ B @ P ) )
% 0.17/0.38 = ( collect @ A @ Q ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % vimage_Collect
% 0.17/0.38 thf(fact_210_finite__vimageD,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,H: A > B,F5: set @ B] :
% 0.17/0.38 ( ( finite_finite2 @ A @ ( vimage @ A @ B @ H @ F5 ) )
% 0.17/0.38 => ( ( ( image @ A @ B @ H @ ( top_top @ ( set @ A ) ) )
% 0.17/0.38 = ( top_top @ ( set @ B ) ) )
% 0.17/0.38 => ( finite_finite2 @ B @ F5 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % finite_vimageD
% 0.17/0.38 thf(fact_211_vimage__subsetD,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,F4: B > A,B6: set @ A,A6: set @ B] :
% 0.17/0.38 ( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 0.17/0.38 = ( top_top @ ( set @ A ) ) )
% 0.17/0.38 => ( ( ord_less_eq @ ( set @ B ) @ ( vimage @ B @ A @ F4 @ B6 ) @ A6 )
% 0.17/0.38 => ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F4 @ A6 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % vimage_subsetD
% 0.17/0.38 thf(fact_212_image__vimage__eq,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,F4: B > A,A6: set @ A] :
% 0.17/0.38 ( ( image @ B @ A @ F4 @ ( vimage @ B @ A @ F4 @ A6 ) )
% 0.17/0.38 = ( inf_inf @ ( set @ A ) @ A6 @ ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % image_vimage_eq
% 0.17/0.38 thf(fact_213_vimage__subsetI,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,F4: A > B,B6: set @ B,A6: set @ A] :
% 0.17/0.38 ( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.38 => ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( image @ A @ B @ F4 @ A6 ) )
% 0.17/0.38 => ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F4 @ B6 ) @ A6 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % vimage_subsetI
% 0.17/0.38 thf(fact_214_Int__iff,axiom,
% 0.17/0.38 ! [A: $tType,C2: A,A6: set @ A,B6: set @ A] :
% 0.17/0.38 ( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) )
% 0.17/0.38 = ( ( member @ A @ C2 @ A6 )
% 0.17/0.38 & ( member @ A @ C2 @ B6 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % Int_iff
% 0.17/0.38 thf(fact_215_IntI,axiom,
% 0.17/0.38 ! [A: $tType,C2: A,A6: set @ A,B6: set @ A] :
% 0.17/0.38 ( ( member @ A @ C2 @ A6 )
% 0.17/0.38 => ( ( member @ A @ C2 @ B6 )
% 0.17/0.38 => ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % IntI
% 0.17/0.38 thf(fact_216_finite__Int,axiom,
% 0.17/0.38 ! [A: $tType,F5: set @ A,G5: set @ A] :
% 0.17/0.38 ( ( ( finite_finite2 @ A @ F5 )
% 0.17/0.38 | ( finite_finite2 @ A @ G5 ) )
% 0.17/0.38 => ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ F5 @ G5 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % finite_Int
% 0.17/0.38 thf(fact_217_Int__subset__iff,axiom,
% 0.17/0.38 ! [A: $tType,C4: set @ A,A6: set @ A,B6: set @ A] :
% 0.17/0.38 ( ( ord_less_eq @ ( set @ A ) @ C4 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) )
% 0.17/0.38 = ( ( ord_less_eq @ ( set @ A ) @ C4 @ A6 )
% 0.17/0.38 & ( ord_less_eq @ ( set @ A ) @ C4 @ B6 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % Int_subset_iff
% 0.17/0.38 thf(fact_218_Int__UNIV,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 0.17/0.38 ( ( ( inf_inf @ ( set @ A ) @ A6 @ B6 )
% 0.17/0.38 = ( top_top @ ( set @ A ) ) )
% 0.17/0.38 = ( ( A6
% 0.17/0.38 = ( top_top @ ( set @ A ) ) )
% 0.17/0.38 & ( B6
% 0.17/0.38 = ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % Int_UNIV
% 0.17/0.38 thf(fact_219_vimage__Int,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,F4: A > B,A6: set @ B,B6: set @ B] :
% 0.17/0.38 ( ( vimage @ A @ B @ F4 @ ( inf_inf @ ( set @ B ) @ A6 @ B6 ) )
% 0.17/0.38 = ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F4 @ A6 ) @ ( vimage @ A @ B @ F4 @ B6 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % vimage_Int
% 0.17/0.38 thf(fact_220_Pow__Int__eq,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 0.17/0.38 ( ( pow @ A @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) )
% 0.17/0.38 = ( inf_inf @ ( set @ ( set @ A ) ) @ ( pow @ A @ A6 ) @ ( pow @ A @ B6 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % Pow_Int_eq
% 0.17/0.38 thf(fact_221_Int__mono,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A,C4: set @ A,B6: set @ A,D3: set @ A] :
% 0.17/0.38 ( ( ord_less_eq @ ( set @ A ) @ A6 @ C4 )
% 0.17/0.38 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ D3 )
% 0.17/0.38 => ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) @ ( inf_inf @ ( set @ A ) @ C4 @ D3 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % Int_mono
% 0.17/0.38 thf(fact_222_Int__lower1,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A,B6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) @ A6 ) ).
% 0.17/0.38
% 0.17/0.38 % Int_lower1
% 0.17/0.38 thf(fact_223_Int__lower2,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A,B6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) @ B6 ) ).
% 0.17/0.38
% 0.17/0.38 % Int_lower2
% 0.17/0.38 thf(fact_224_Int__absorb1,axiom,
% 0.17/0.38 ! [A: $tType,B6: set @ A,A6: set @ A] :
% 0.17/0.38 ( ( ord_less_eq @ ( set @ A ) @ B6 @ A6 )
% 0.17/0.38 => ( ( inf_inf @ ( set @ A ) @ A6 @ B6 )
% 0.17/0.38 = B6 ) ) ).
% 0.17/0.38
% 0.17/0.38 % Int_absorb1
% 0.17/0.38 thf(fact_225_Int__absorb2,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 0.17/0.38 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.17/0.38 => ( ( inf_inf @ ( set @ A ) @ A6 @ B6 )
% 0.17/0.38 = A6 ) ) ).
% 0.17/0.38
% 0.17/0.38 % Int_absorb2
% 0.17/0.38 thf(fact_226_Int__greatest,axiom,
% 0.17/0.38 ! [A: $tType,C4: set @ A,A6: set @ A,B6: set @ A] :
% 0.17/0.38 ( ( ord_less_eq @ ( set @ A ) @ C4 @ A6 )
% 0.17/0.38 => ( ( ord_less_eq @ ( set @ A ) @ C4 @ B6 )
% 0.17/0.38 => ( ord_less_eq @ ( set @ A ) @ C4 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % Int_greatest
% 0.17/0.38 thf(fact_227_Int__Collect__mono,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A,B6: set @ A,P: A > $o,Q: A > $o] :
% 0.17/0.38 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.17/0.38 => ( ! [X3: A] :
% 0.17/0.38 ( ( member @ A @ X3 @ A6 )
% 0.17/0.38 => ( ( P @ X3 )
% 0.17/0.38 => ( Q @ X3 ) ) )
% 0.17/0.38 => ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ ( collect @ A @ P ) ) @ ( inf_inf @ ( set @ A ) @ B6 @ ( collect @ A @ Q ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % Int_Collect_mono
% 0.17/0.38 thf(fact_228_inj__on__image__Int,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,F4: A > B,C4: set @ A,A6: set @ A,B6: set @ A] :
% 0.17/0.38 ( ( inj_on @ A @ B @ F4 @ C4 )
% 0.17/0.38 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ C4 )
% 0.17/0.38 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
% 0.17/0.38 => ( ( image @ A @ B @ F4 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) )
% 0.17/0.38 = ( inf_inf @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) @ ( image @ A @ B @ F4 @ B6 ) ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % inj_on_image_Int
% 0.17/0.38 thf(fact_229_inj__on__image__eq__iff,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,F4: A > B,C4: set @ A,A6: set @ A,B6: set @ A] :
% 0.17/0.38 ( ( inj_on @ A @ B @ F4 @ C4 )
% 0.17/0.38 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ C4 )
% 0.17/0.38 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
% 0.17/0.38 => ( ( ( image @ A @ B @ F4 @ A6 )
% 0.17/0.38 = ( image @ A @ B @ F4 @ B6 ) )
% 0.17/0.38 = ( A6 = B6 ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % inj_on_image_eq_iff
% 0.17/0.38 thf(fact_230_inj__on__image__mem__iff,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,F4: A > B,B6: set @ A,A3: A,A6: set @ A] :
% 0.17/0.38 ( ( inj_on @ A @ B @ F4 @ B6 )
% 0.17/0.38 => ( ( member @ A @ A3 @ B6 )
% 0.17/0.38 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.17/0.38 => ( ( member @ B @ ( F4 @ A3 ) @ ( image @ A @ B @ F4 @ A6 ) )
% 0.17/0.38 = ( member @ A @ A3 @ A6 ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % inj_on_image_mem_iff
% 0.17/0.38 thf(fact_231_inj__on__image__mem__iff__alt,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,F4: A > B,B6: set @ A,A6: set @ A,A3: A] :
% 0.17/0.38 ( ( inj_on @ A @ B @ F4 @ B6 )
% 0.17/0.38 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 0.17/0.38 => ( ( member @ B @ ( F4 @ A3 ) @ ( image @ A @ B @ F4 @ A6 ) )
% 0.17/0.38 => ( ( member @ A @ A3 @ B6 )
% 0.17/0.38 => ( member @ A @ A3 @ A6 ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % inj_on_image_mem_iff_alt
% 0.17/0.38 thf(fact_232_image__Int__subset,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B,B6: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F4 @ ( inf_inf @ ( set @ B ) @ A6 @ B6 ) ) @ ( inf_inf @ ( set @ A ) @ ( image @ B @ A @ F4 @ A6 ) @ ( image @ B @ A @ F4 @ B6 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % image_Int_subset
% 0.17/0.38 thf(fact_233_vimage__inter__cong,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,S4: set @ A,F4: A > B,G3: A > B,Y: set @ B] :
% 0.17/0.38 ( ! [W: A] :
% 0.17/0.38 ( ( member @ A @ W @ S4 )
% 0.17/0.38 => ( ( F4 @ W )
% 0.17/0.38 = ( G3 @ W ) ) )
% 0.17/0.38 => ( ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F4 @ Y ) @ S4 )
% 0.17/0.38 = ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ G3 @ Y ) @ S4 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % vimage_inter_cong
% 0.17/0.38 thf(fact_234_finite__vimage__IntI,axiom,
% 0.17/0.38 ! [A: $tType,B: $tType,F5: set @ A,H: B > A,A6: set @ B] :
% 0.17/0.38 ( ( finite_finite2 @ A @ F5 )
% 0.17/0.38 => ( ( inj_on @ B @ A @ H @ A6 )
% 0.17/0.38 => ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H @ F5 ) @ A6 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % finite_vimage_IntI
% 0.17/0.38 thf(fact_235_image__Int,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,F4: A > B,A6: set @ A,B6: set @ A] :
% 0.17/0.38 ( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.38 => ( ( image @ A @ B @ F4 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) )
% 0.17/0.38 = ( inf_inf @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) @ ( image @ A @ B @ F4 @ B6 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % image_Int
% 0.17/0.38 thf(fact_236_inj__on__image__iff,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,A6: set @ A,G3: A > B,F4: A > A] :
% 0.17/0.38 ( ! [X3: A] :
% 0.17/0.38 ( ( member @ A @ X3 @ A6 )
% 0.17/0.38 => ! [Xa: A] :
% 0.17/0.38 ( ( member @ A @ Xa @ A6 )
% 0.17/0.38 => ( ( ( G3 @ ( F4 @ X3 ) )
% 0.17/0.38 = ( G3 @ ( F4 @ Xa ) ) )
% 0.17/0.38 = ( ( G3 @ X3 )
% 0.17/0.38 = ( G3 @ Xa ) ) ) ) )
% 0.17/0.38 => ( ( inj_on @ A @ A @ F4 @ A6 )
% 0.17/0.38 => ( ( inj_on @ A @ B @ G3 @ ( image @ A @ A @ F4 @ A6 ) )
% 0.17/0.38 = ( inj_on @ A @ B @ G3 @ A6 ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % inj_on_image_iff
% 0.17/0.38 thf(fact_237_inj__swap,axiom,
% 0.17/0.38 ! [B: $tType,A: $tType,A6: set @ ( product_prod @ A @ B )] : ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) @ ( product_swap @ A @ B ) @ A6 ) ).
% 0.17/0.38
% 0.17/0.38 % inj_swap
% 0.17/0.38 thf(fact_238_Int__left__commute,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A,B6: set @ A,C4: set @ A] :
% 0.17/0.38 ( ( inf_inf @ ( set @ A ) @ A6 @ ( inf_inf @ ( set @ A ) @ B6 @ C4 ) )
% 0.17/0.38 = ( inf_inf @ ( set @ A ) @ B6 @ ( inf_inf @ ( set @ A ) @ A6 @ C4 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % Int_left_commute
% 0.17/0.38 thf(fact_239_Int__left__absorb,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 0.17/0.38 ( ( inf_inf @ ( set @ A ) @ A6 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) )
% 0.17/0.38 = ( inf_inf @ ( set @ A ) @ A6 @ B6 ) ) ).
% 0.17/0.38
% 0.17/0.38 % Int_left_absorb
% 0.17/0.38 thf(fact_240_Int__commute,axiom,
% 0.17/0.38 ! [A: $tType] :
% 0.17/0.38 ( ( inf_inf @ ( set @ A ) )
% 0.17/0.38 = ( ^ [A7: set @ A,B7: set @ A] : ( inf_inf @ ( set @ A ) @ B7 @ A7 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % Int_commute
% 0.17/0.38 thf(fact_241_Int__absorb,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A] :
% 0.17/0.38 ( ( inf_inf @ ( set @ A ) @ A6 @ A6 )
% 0.17/0.38 = A6 ) ).
% 0.17/0.38
% 0.17/0.38 % Int_absorb
% 0.17/0.38 thf(fact_242_Int__assoc,axiom,
% 0.17/0.38 ! [A: $tType,A6: set @ A,B6: set @ A,C4: set @ A] :
% 0.17/0.38 ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) @ C4 )
% 0.17/0.38 = ( inf_inf @ ( set @ A ) @ A6 @ ( inf_inf @ ( set @ A ) @ B6 @ C4 ) ) ) ).
% 0.17/0.38
% 0.17/0.38 % Int_assoc
% 0.17/0.38 thf(fact_243_IntD2,axiom,
% 0.17/0.38 ! [A: $tType,C2: A,A6: set @ A,B6: set @ A] :
% 0.17/0.38 ( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) )
% 0.17/0.38 => ( member @ A @ C2 @ B6 ) ) ).
% 0.17/0.38
% 0.17/0.38 % IntD2
% 0.17/0.38 thf(fact_244_IntD1,axiom,
% 0.17/0.38 ! [A: $tType,C2: A,A6: set @ A,B6: set @ A] :
% 0.17/0.38 ( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) )
% 0.17/0.39 => ( member @ A @ C2 @ A6 ) ) ).
% 0.17/0.39
% 0.17/0.39 % IntD1
% 0.17/0.39 thf(fact_245_IntE,axiom,
% 0.17/0.39 ! [A: $tType,C2: A,A6: set @ A,B6: set @ A] :
% 0.17/0.39 ( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) )
% 0.17/0.39 => ~ ( ( member @ A @ C2 @ A6 )
% 0.17/0.39 => ~ ( member @ A @ C2 @ B6 ) ) ) ).
% 0.17/0.39
% 0.17/0.39 % IntE
% 0.17/0.39 thf(fact_246_Int__UNIV__left,axiom,
% 0.17/0.39 ! [A: $tType,B6: set @ A] :
% 0.17/0.39 ( ( inf_inf @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B6 )
% 0.17/0.39 = B6 ) ).
% 0.17/0.39
% 0.17/0.39 % Int_UNIV_left
% 0.17/0.39 thf(fact_247_Int__UNIV__right,axiom,
% 0.17/0.39 ! [A: $tType,A6: set @ A] :
% 0.17/0.39 ( ( inf_inf @ ( set @ A ) @ A6 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.39 = A6 ) ).
% 0.17/0.39
% 0.17/0.39 % Int_UNIV_right
% 0.17/0.39 thf(fact_248_inj__eq,axiom,
% 0.17/0.39 ! [B: $tType,A: $tType,F4: A > B,X: A,Y: A] :
% 0.17/0.39 ( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.39 => ( ( ( F4 @ X )
% 0.17/0.39 = ( F4 @ Y ) )
% 0.17/0.39 = ( X = Y ) ) ) ).
% 0.17/0.39
% 0.17/0.39 % inj_eq
% 0.17/0.39 thf(fact_249_injI,axiom,
% 0.17/0.39 ! [B: $tType,A: $tType,F4: A > B] :
% 0.17/0.39 ( ! [X3: A,Y3: A] :
% 0.17/0.39 ( ( ( F4 @ X3 )
% 0.17/0.39 = ( F4 @ Y3 ) )
% 0.17/0.39 => ( X3 = Y3 ) )
% 0.17/0.39 => ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 % injI
% 0.17/0.39 thf(fact_250_injD,axiom,
% 0.17/0.39 ! [B: $tType,A: $tType,F4: A > B,X: A,Y: A] :
% 0.17/0.39 ( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.39 => ( ( ( F4 @ X )
% 0.17/0.39 = ( F4 @ Y ) )
% 0.17/0.39 => ( X = Y ) ) ) ).
% 0.17/0.39
% 0.17/0.39 % injD
% 0.17/0.39 thf(fact_251_range__ex1__eq,axiom,
% 0.17/0.39 ! [B: $tType,A: $tType,F4: A > B,B3: B] :
% 0.17/0.39 ( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.39 => ( ( member @ B @ B3 @ ( image @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) ) )
% 0.17/0.39 = ( ? [X4: A] :
% 0.17/0.39 ( ( B3
% 0.17/0.39 = ( F4 @ X4 ) )
% 0.17/0.39 & ! [Y5: A] :
% 0.17/0.39 ( ( B3
% 0.17/0.39 = ( F4 @ Y5 ) )
% 0.17/0.39 => ( Y5 = X4 ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 % range_ex1_eq
% 0.17/0.39 thf(fact_252_inj__image__eq__iff,axiom,
% 0.17/0.39 ! [B: $tType,A: $tType,F4: A > B,A6: set @ A,B6: set @ A] :
% 0.17/0.39 ( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.39 => ( ( ( image @ A @ B @ F4 @ A6 )
% 0.17/0.39 = ( image @ A @ B @ F4 @ B6 ) )
% 0.17/0.39 = ( A6 = B6 ) ) ) ).
% 0.17/0.39
% 0.17/0.39 % inj_image_eq_iff
% 0.17/0.39 thf(fact_253_inj__image__mem__iff,axiom,
% 0.17/0.39 ! [B: $tType,A: $tType,F4: A > B,A3: A,A6: set @ A] :
% 0.17/0.39 ( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 0.17/0.39 => ( ( member @ B @ ( F4 @ A3 ) @ ( image @ A @ B @ F4 @ A6 ) )
% 0.17/0.39 = ( member @ A @ A3 @ A6 ) ) ) ).
% 0.17/0.39
% 0.17/0.39 % inj_image_mem_iff
% 0.17/0.39 thf(fact_254_finite__imageD,axiom,
% 0.17/0.39 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B] :
% 0.17/0.39 ( ( finite_finite2 @ A @ ( image @ B @ A @ F4 @ A6 ) )
% 0.17/0.39 => ( ( inj_on @ B @ A @ F4 @ A6 )
% 0.17/0.39 => ( finite_finite2 @ B @ A6 ) ) ) ).
% 0.17/0.39
% 0.17/0.39 % finite_imageD
% 0.17/0.39 thf(fact_255_finite__image__iff,axiom,
% 0.17/0.39 ! [B: $tType,A: $tType,F4: A > B,A6: set @ A] :
% 0.17/0.39 ( ( inj_on @ A @ B @ F4 @ A6 )
% 0.17/0.39 => ( ( finite_finite2 @ B @ ( image @ A @ B @ F4 @ A6 ) )
% 0.17/0.39 = ( finite_finite2 @ A @ A6 ) ) ) ).
% 0.17/0.39
% 0.17/0.39 % finite_image_iff
% 0.17/0.39
% 0.17/0.39 %----Type constructors (15)
% 0.17/0.39 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 0.17/0.39 ! [A8: $tType,A9: $tType] :
% 0.17/0.39 ( ( order_top @ A9 @ ( type2 @ A9 ) )
% 0.17/0.39 => ( order_top @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.17/0.39 ! [A8: $tType,A9: $tType] :
% 0.17/0.39 ( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 0.17/0.39 & ( finite_finite @ A9 @ ( type2 @ A9 ) ) )
% 0.17/0.39 => ( finite_finite @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(tcon_fun___Orderings_Otop,axiom,
% 0.17/0.39 ! [A8: $tType,A9: $tType] :
% 0.17/0.39 ( ( top @ A9 @ ( type2 @ A9 ) )
% 0.17/0.39 => ( top @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(tcon_fun___Orderings_Oord,axiom,
% 0.17/0.39 ! [A8: $tType,A9: $tType] :
% 0.17/0.39 ( ( ord @ A9 @ ( type2 @ A9 ) )
% 0.17/0.39 => ( ord @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(tcon_Set_Oset___Orderings_Oorder__top_1,axiom,
% 0.17/0.39 ! [A8: $tType] : ( order_top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(tcon_Set_Oset___Finite__Set_Ofinite_2,axiom,
% 0.17/0.39 ! [A8: $tType] :
% 0.17/0.39 ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 0.17/0.39 => ( finite_finite @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(tcon_Set_Oset___Orderings_Otop_3,axiom,
% 0.17/0.39 ! [A8: $tType] : ( top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(tcon_Set_Oset___Orderings_Oord_4,axiom,
% 0.17/0.39 ! [A8: $tType] : ( ord @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(tcon_HOL_Obool___Orderings_Oorder__top_5,axiom,
% 0.17/0.39 order_top @ $o @ ( type2 @ $o ) ).
% 0.17/0.39
% 0.17/0.39 thf(tcon_HOL_Obool___Finite__Set_Ofinite_6,axiom,
% 0.17/0.39 finite_finite @ $o @ ( type2 @ $o ) ).
% 0.17/0.39
% 0.17/0.39 thf(tcon_HOL_Obool___Orderings_Otop_7,axiom,
% 0.17/0.39 top @ $o @ ( type2 @ $o ) ).
% 0.17/0.39
% 0.17/0.39 thf(tcon_HOL_Obool___Orderings_Oord_8,axiom,
% 0.17/0.39 ord @ $o @ ( type2 @ $o ) ).
% 0.17/0.39
% 0.17/0.39 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_9,axiom,
% 0.17/0.39 ! [A8: $tType,A9: $tType] :
% 0.17/0.39 ( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 0.17/0.39 & ( finite_finite @ A9 @ ( type2 @ A9 ) ) )
% 0.17/0.39 => ( finite_finite @ ( sum_sum @ A8 @ A9 ) @ ( type2 @ ( sum_sum @ A8 @ A9 ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(tcon_Option_Ooption___Finite__Set_Ofinite_10,axiom,
% 0.17/0.39 ! [A8: $tType] :
% 0.17/0.39 ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 0.17/0.39 => ( finite_finite @ ( option @ A8 ) @ ( type2 @ ( option @ A8 ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_11,axiom,
% 0.17/0.39 ! [A8: $tType,A9: $tType] :
% 0.17/0.39 ( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 0.17/0.39 & ( finite_finite @ A9 @ ( type2 @ A9 ) ) )
% 0.17/0.39 => ( finite_finite @ ( product_prod @ A8 @ A9 ) @ ( type2 @ ( product_prod @ A8 @ A9 ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 %----Conjectures (1)
% 0.17/0.39 thf(conj_0,conjecture,
% 0.17/0.39 ( ( root @ ( unfold @ ( product_prod @ dtree @ dtree ) @ parall1914194362_par_r @ parall1914194347_par_c @ ( product_Pair @ dtree @ dtree @ tr1 @ tr2 ) ) )
% 0.17/0.39 = ( parall1518086719_Nplus @ ( root @ tr1 ) @ ( root @ tr2 ) ) ) ).
% 0.17/0.39
% 0.17/0.39 %------------------------------------------------------------------------------
% 0.17/0.39 ------------------------
% 3.10/1.61 ---- Embedded in THF ---
% 3.10/1.61 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.10/1.61 %%% Generated on Mon Feb 23 23:16:42 EST 2026
% 3.10/1.61 %%% using '$$dhol' embedding, version 1.3.0.
% 3.10/1.61 %%% Logic specification used:
% 3.10/1.61 %%% thf(spec, logic, $$dhol).
% 3.10/1.61
% 3.10/1.61 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.qb6mFvvipo/DTF2DTF20809.p
% 3.10/1.61 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 3.10/1.61 thf(ty_t_Sum__Type_Osum, type, sum_sum: ($tType > ($tType > $tType))).
% 3.10/1.61 thf(ty_t_Option_Ooption, type, option: ($tType > $tType)).
% 3.10/1.61 thf(ty_t_DTree_Odtree, type, dtree: $tType).
% 3.10/1.61 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 3.10/1.61 thf(ty_t_DTree_OT, type, t: $tType).
% 3.10/1.61 thf(ty_t_DTree_ON, type, n: $tType).
% 3.10/1.61 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 3.10/1.61 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.10/1.61 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.10/1.61 thf(sy_cl_Orderings_Otop, type, top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.10/1.61 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.10/1.61 thf(sy_cl_Nat_Osemiring__char__0, type, semiring_char_0: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.10/1.61 thf(sy_cl_Orderings_Oorder__top, type, order_top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 3.10/1.61 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain, type, bNF_Ca1785829860lChain: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((A > B) > $o))))).
% 3.10/1.61 thf(sy_c_BNF__Def_OfstOp, type, bNF_fstOp: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > $o)) > ((B > (C > $o)) > (((product_prod @ A) @ C) > ((product_prod @ A) @ B))))))).
% 3.10/1.61 thf(sy_c_BNF__Def_Opick__middlep, type, bNF_pick_middlep: (!> [B:$tType,A:$tType,C:$tType]: (((B > (A > $o)) > ((A > (C > $o)) > (B > (C > A))))))).
% 3.10/1.61 thf(sy_c_BNF__Def_OsndOp, type, bNF_sndOp: (!> [C:$tType,A:$tType,B:$tType]: (((C > (A > $o)) > ((A > (B > $o)) > (((product_prod @ C) @ B) > ((product_prod @ A) @ B))))))).
% 3.10/1.61 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2, type, bNF_Greatest_image2: (!> [C:$tType,A:$tType,B:$tType]: (((set @ C) > ((C > A) > ((C > B) > (set @ ((product_prod @ A) @ B)))))))).
% 3.10/1.61 thf(sy_c_DTree_ONode, type, node: (n > ((set @ ((sum_sum @ t) @ dtree)) > dtree))).
% 3.10/1.61 thf(sy_c_DTree_Ocont, type, cont: (dtree > (set @ ((sum_sum @ t) @ dtree)))).
% 3.10/1.61 thf(sy_c_DTree_Ocorec, type, corec: (!> [A:$tType]: (((A > n) > ((A > (set @ ((sum_sum @ t) @ ((sum_sum @ dtree) @ A)))) > (A > dtree)))))).
% 3.10/1.61 thf(sy_c_DTree_Odtree_Oroot, type, root: (dtree > n)).
% 3.10/1.61 thf(sy_c_DTree_Ounfold, type, unfold: (!> [A:$tType]: (((A > n) > ((A > (set @ ((sum_sum @ t) @ A))) > (A > dtree)))))).
% 3.10/1.61 thf(sy_c_Finite__Set_Ofinite, type, finite_finite2: (!> [A:$tType]: (((set @ A) > $o)))).
% 3.10/1.61 thf(sy_c_Fun_Oinj__on, type, inj_on: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > $o))))).
% 3.10/1.61 thf(sy_c_Fun_Oswap, type, swap: (!> [A:$tType,B:$tType]: ((A > (A > ((A > B) > (A > B))))))).
% 3.10/1.61 thf(sy_c_Lattices_Oinf__class_Oinf, type, inf_inf: (!> [A:$tType]: ((A > (A > A))))).
% 3.10/1.61 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 3.10/1.61 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 3.10/1.61 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_ONplus, type, parall1518086719_Nplus: (n > (n > n))).
% 3.10/1.61 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar, type, parall1899940088le_par: (((product_prod @ dtree) @ dtree) > dtree)).
% 3.10/1.61 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__c, type, parall1914194347_par_c: (((product_prod @ dtree) @ dtree) > (set @ ((sum_sum @ t) @ ((product_prod @ dtree) @ dtree))))).
% 3.10/1.61 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__r, type, parall1914194362_par_r: (((product_prod @ dtree) @ dtree) > n)).
% 3.10/1.61 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__r__rel, type, parall556292031_r_rel: (((product_prod @ dtree) @ dtree) > (((product_prod @ dtree) @ dtree) > $o))).
% 3.10/1.61 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 3.10/1.61 thf(sy_c_Product__Type_Ocurry, type, product_curry: (!> [A:$tType,B:$tType,C:$tType]: (((((product_prod @ A) @ B) > C) > (A > (B > C)))))).
% 3.10/1.61 thf(sy_c_Product__Type_Ointernal__case__prod, type, produc2004651681e_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 3.10/1.61 thf(sy_c_Product__Type_Omap__prod, type, product_map_prod: (!> [A:$tType,C:$tType,B:$tType,D:$tType]: (((A > C) > ((B > D) > (((product_prod @ A) @ B) > ((product_prod @ C) @ D))))))).
% 3.10/1.61 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 3.10/1.61 thf(sy_c_Product__Type_Oprod_Ocase__prod, type, product_case_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 3.10/1.61 thf(sy_c_Product__Type_Oprod_Ofst, type, product_fst: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > A)))).
% 3.10/1.61 thf(sy_c_Product__Type_Oprod_Osnd, type, product_snd: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > B)))).
% 3.10/1.61 thf(sy_c_Product__Type_Oprod_Oswap, type, product_swap: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > ((product_prod @ B) @ A))))).
% 3.10/1.61 thf(sy_c_Product__Type_Oscomp, type, product_scomp: (!> [A:$tType,B:$tType,C:$tType,D:$tType]: (((A > ((product_prod @ B) @ C)) > ((B > (C > D)) > (A > D)))))).
% 3.10/1.61 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 3.10/1.61 thf(sy_c_Relation_Oinv__image, type, inv_image: (!> [B:$tType,A:$tType]: (((set @ ((product_prod @ B) @ B)) > ((A > B) > (set @ ((product_prod @ A) @ A))))))).
% 3.10/1.61 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 3.10/1.61 thf(sy_c_Set_OPow, type, pow: (!> [A:$tType]: (((set @ A) > (set @ (set @ A)))))).
% 3.10/1.61 thf(sy_c_Set_Oimage, type, image: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))).
% 3.10/1.61 thf(sy_c_Set_Ovimage, type, vimage: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ B) > (set @ A)))))).
% 3.10/1.61 thf(sy_c_Wellfounded_Oaccp, type, accp: (!> [A:$tType]: (((A > (A > $o)) > (A > $o))))).
% 3.10/1.61 thf(sy_c_Wellfounded_Olex__prod, type, lex_prod: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ ((product_prod @ B) @ B)) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))))))).
% 3.10/1.61 thf(sy_c_Wfrec_Osame__fst, type, same_fst: (!> [A:$tType,B:$tType]: (((A > $o) > ((A > (set @ ((product_prod @ B) @ B))) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))))))).
% 3.10/1.61 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 3.10/1.61 thf(sy_v_tr1, type, tr1: dtree).
% 3.10/1.61 thf(sy_v_tr2, type, tr2: dtree).
% 3.10/1.61 thf(fact_0__092_060open_062root_A_Iunfold_Apar__r_Apar__c_A_Itr1_M_Atr2_J_J_A_061_Apar__r_A_Itr1_M_Atr2_J_092_060close_062, axiom, ((root @ ((((unfold @ ((product_prod @ dtree) @ dtree)) @ parall1914194362_par_r) @ parall1914194347_par_c) @ ((((product_Pair @ dtree) @ dtree) @ tr1) @ tr2))) = (parall1914194362_par_r @ ((((product_Pair @ dtree) @ dtree) @ tr1) @ tr2)))).
% 3.10/1.61 thf(fact_1_Nplus__comm, axiom, (parall1518086719_Nplus = (^ [A2:n,B2:n]: (((parall1518086719_Nplus @ B2) @ A2))))).
% 3.10/1.61 thf(fact_2_Nplus__assoc, axiom, (! [A3:n,B3:n,C2:n]: ((((parall1518086719_Nplus @ ((parall1518086719_Nplus @ A3) @ B3)) @ C2) = ((parall1518086719_Nplus @ A3) @ ((parall1518086719_Nplus @ B3) @ C2)))))).
% 3.10/1.61 thf(fact_3_par__r_Ocases, axiom, (! [X:((product_prod @ dtree) @ dtree)]: ((~ ((! [Tr1:dtree,Tr2:dtree]: ((X != ((((product_Pair @ dtree) @ dtree) @ Tr1) @ Tr2))))))))).
% 3.10/1.61 thf(fact_4_par__r_Oelims, axiom, (! [X:((product_prod @ dtree) @ dtree),Y:n]: ((((parall1914194362_par_r @ X) = Y) => (~ ((! [Tr1:dtree,Tr2:dtree]: (((X = ((((product_Pair @ dtree) @ dtree) @ Tr1) @ Tr2)) => (Y != ((parall1518086719_Nplus @ (root @ Tr1)) @ (root @ Tr2)))))))))))).
% 3.10/1.61 thf(fact_5_par__r_Osimps, axiom, (! [Tr12:dtree,Tr22:dtree]: (((parall1914194362_par_r @ ((((product_Pair @ dtree) @ dtree) @ Tr12) @ Tr22)) = ((parall1518086719_Nplus @ (root @ Tr12)) @ (root @ Tr22)))))).
% 3.10/1.61 thf(fact_6_par__r_Oinduct, axiom, (! [P:(((product_prod @ dtree) @ dtree) > $o),A0:((product_prod @ dtree) @ dtree)]: (((! [Tr1:dtree,Tr2:dtree]: ((P @ ((((product_Pair @ dtree) @ dtree) @ Tr1) @ Tr2)))) => (P @ A0))))).
% 3.10/1.61 thf(fact_7_par__def, axiom, (parall1899940088le_par = (((unfold @ ((product_prod @ dtree) @ dtree)) @ parall1914194362_par_r) @ parall1914194347_par_c))).
% 3.10/1.61 thf(fact_8_unfold_I1_J, axiom, (! [A:$tType,Rt:(A > n),Ct:(A > (set @ ((sum_sum @ t) @ A))),B3:A]: (((root @ ((((unfold @ A) @ Rt) @ Ct) @ B3)) = (Rt @ B3))))).
% 3.10/1.61 thf(fact_9_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X2:B,Y1:A,Y2:B]: (((((((product_Pair @ A) @ B) @ X1) @ X2) = ((((product_Pair @ A) @ B) @ Y1) @ Y2)) = ((X1 = Y1) & (X2 = Y2)))))).
% 3.10/1.61 thf(fact_10_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A3:A,B3:B,A4:A,B4:B]: (((((((product_Pair @ A) @ B) @ A3) @ B3) = ((((product_Pair @ A) @ B) @ A4) @ B4)) = ((A3 = A4) & (B3 = B4)))))).
% 3.10/1.61 thf(fact_11_par__r_Opelims, axiom, (! [X:((product_prod @ dtree) @ dtree),Y:n]: ((((parall1914194362_par_r @ X) = Y) => ((((accp @ ((product_prod @ dtree) @ dtree)) @ parall556292031_r_rel) @ X) => (~ ((! [Tr1:dtree,Tr2:dtree]: (((X = ((((product_Pair @ dtree) @ dtree) @ Tr1) @ Tr2)) => ((Y = ((parall1518086719_Nplus @ (root @ Tr1)) @ (root @ Tr2))) => (~ ((((accp @ ((product_prod @ dtree) @ dtree)) @ parall556292031_r_rel) @ ((((product_Pair @ dtree) @ dtree) @ Tr1) @ Tr2))))))))))))))).
% 3.10/1.61 thf(fact_12_surj__pair, axiom, (! [A:$tType,B:$tType,P2:((product_prod @ A) @ B)]: ((? [X3:A,Y3:B]: ((P2 = ((((product_Pair @ A) @ B) @ X3) @ Y3))))))).
% 3.10/1.61 thf(fact_13_prod__cases, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),P2:((product_prod @ A) @ B)]: (((! [A5:A,B5:B]: ((P @ ((((product_Pair @ A) @ B) @ A5) @ B5)))) => (P @ P2))))).
% 3.10/1.61 thf(fact_14_Pair__inject, axiom, (! [A:$tType,B:$tType,A3:A,B3:B,A4:A,B4:B]: (((((((product_Pair @ A) @ B) @ A3) @ B3) = ((((product_Pair @ A) @ B) @ A4) @ B4)) => (~ (((A3 = A4) => (B3 != B4)))))))).
% 3.10/1.61 thf(fact_15_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A5:A,B5:B,C3:C]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B5) @ C3)))))))))).
% 3.10/1.61 thf(fact_16_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: ((~ ((! [A5:A,B5:B,C3:C,D2:D]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B5) @ ((((product_Pair @ C) @ D) @ C3) @ D2))))))))))).
% 3.10/1.61 thf(fact_17_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: ((~ ((! [A5:A,B5:B,C3:C,D2:D,E2:E]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C3) @ ((((product_Pair @ D) @ E) @ D2) @ E2)))))))))))).
% 3.10/1.61 thf(fact_18_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))))]: ((~ ((! [A5:A,B5:B,C3:C,D2:D,E2:E,F2:F]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F)) @ D2) @ ((((product_Pair @ E) @ F) @ E2) @ F2))))))))))))).
% 3.10/1.61 thf(fact_19_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A5:A,B5:B]: ((P @ ((((product_Pair @ A) @ B) @ A5) @ B5)))) => (P @ Prod))))).
% 3.10/1.61 thf(fact_20_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y:((product_prod @ A) @ B)]: ((~ ((! [A5:A,B5:B]: ((Y != ((((product_Pair @ A) @ B) @ A5) @ B5))))))))).
% 3.10/1.61 thf(fact_21_prod__induct7, axiom, (! [G:$tType,F:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))))]: (((! [A5:A,B5:B,C3:C,D2:D,E2:E,F2:F,G2:G]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F) @ G)) @ E2) @ ((((product_Pair @ F) @ G) @ F2) @ G2))))))))) => (P @ X))))).
% 3.10/1.61 thf(fact_22_prod__induct6, axiom, (! [F:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))))]: (((! [A5:A,B5:B,C3:C,D2:D,E2:E,F2:F]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F)) @ D2) @ ((((product_Pair @ E) @ F) @ E2) @ F2)))))))) => (P @ X))))).
% 3.10/1.61 thf(fact_23_prod__induct5, axiom, (! [E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: (((! [A5:A,B5:B,C3:C,D2:D,E2:E]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C3) @ ((((product_Pair @ D) @ E) @ D2) @ E2))))))) => (P @ X))))).
% 3.10/1.61 thf(fact_24_prod__induct4, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: (((! [A5:A,B5:B,C3:C,D2:D]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B5) @ ((((product_Pair @ C) @ D) @ C3) @ D2)))))) => (P @ X))))).
% 3.10/1.61 thf(fact_25_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A5:A,B5:B,C3:C]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B5) @ C3))))) => (P @ X))))).
% 3.10/1.61 thf(fact_26_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F:$tType,G:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))))]: ((~ ((! [A5:A,B5:B,C3:C,D2:D,E2:E,F2:F,G2:G]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))) @ C3) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F) @ G)) @ E2) @ ((((product_Pair @ F) @ G) @ F2) @ G2)))))))))))))).
% 3.10/1.61 thf(fact_27_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A3:A,B3:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) = ((F1 @ A3) @ B3))))).
% 3.10/1.61 thf(fact_28_internal__case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,C2:(B > (C > A)),A3:B,B3:C]: (((((((produc2004651681e_prod @ B) @ C) @ A) @ C2) @ ((((product_Pair @ B) @ C) @ A3) @ B3)) = ((C2 @ A3) @ B3))))).
% 3.10/1.61 thf(fact_29_root__Node, axiom, (! [N:n,As:(set @ ((sum_sum @ t) @ dtree))]: (((root @ ((node @ N) @ As)) = N)))).
% 3.10/1.61 thf(fact_30_accp_Ocases, axiom, (! [A:$tType,R:(A > (A > $o)),A3:A]: (((((accp @ A) @ R) @ A3) => (! [Y4:A]: ((((R @ Y4) @ A3) => (((accp @ A) @ R) @ Y4)))))))).
% 3.10/1.61 thf(fact_31_accp_Osimps, axiom, (! [A:$tType]: (((accp @ A) = (^ [R2:(A > (A > $o)),A2:A]: ((? [X4:A]: (((A2 = X4) & (! [Y5:A]: ((((R2 @ Y5) @ X4) => (((accp @ A) @ R2) @ Y5))))))))))))).
% 3.10/1.61 thf(fact_32_accp_Ointros, axiom, (! [A:$tType,R:(A > (A > $o)),X:A]: (((! [Y3:A]: ((((R @ Y3) @ X) => (((accp @ A) @ R) @ Y3)))) => (((accp @ A) @ R) @ X))))).
% 3.10/1.61 thf(fact_33_accp__induct, axiom, (! [A:$tType,R:(A > (A > $o)),A3:A,P:(A > $o)]: (((((accp @ A) @ R) @ A3) => ((! [X3:A]: (((((accp @ A) @ R) @ X3) => ((! [Y4:A]: ((((R @ Y4) @ X3) => (P @ Y4)))) => (P @ X3))))) => (P @ A3)))))).
% 3.10/1.61 thf(fact_34_accp_Oinducts, axiom, (! [A:$tType,R:(A > (A > $o)),X:A,P:(A > $o)]: (((((accp @ A) @ R) @ X) => ((! [X3:A]: (((! [Y4:A]: ((((R @ Y4) @ X3) => (((accp @ A) @ R) @ Y4)))) => ((! [Y4:A]: ((((R @ Y4) @ X3) => (P @ Y4)))) => (P @ X3))))) => (P @ X)))))).
% 3.10/1.61 thf(fact_35_accp__downward, axiom, (! [A:$tType,R:(A > (A > $o)),B3:A,A3:A]: (((((accp @ A) @ R) @ B3) => (((R @ A3) @ B3) => (((accp @ A) @ R) @ A3)))))).
% 3.10/1.61 thf(fact_36_not__accp__down, axiom, (! [A:$tType,R3:(A > (A > $o)),X:A]: (((~ ((((accp @ A) @ R3) @ X))) => (~ ((! [Z:A]: ((((R3 @ Z) @ X) => (((accp @ A) @ R3) @ Z)))))))))).
% 3.10/1.61 thf(fact_37_accp__induct__rule, axiom, (! [A:$tType,R:(A > (A > $o)),A3:A,P:(A > $o)]: (((((accp @ A) @ R) @ A3) => ((! [X3:A]: (((((accp @ A) @ R) @ X3) => ((! [Y4:A]: ((((R @ Y4) @ X3) => (P @ Y4)))) => (P @ X3))))) => (P @ A3)))))).
% 3.10/1.61 thf(fact_38_in__lex__prod, axiom, (! [A:$tType,B:$tType,A3:A,B3:B,A4:A,B4:B,R:(set @ ((product_prod @ A) @ A)),S:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ ((((product_Pair @ A) @ B) @ A4) @ B4))) @ ((((lex_prod @ A) @ B) @ R) @ S)) = ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ A4)) @ R) | ((A3 = A4) & (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ B3) @ B4)) @ S))))))).
% 3.10/1.61 thf(fact_39_Node__root__cont, axiom, (! [Tr:dtree]: ((((node @ (root @ Tr)) @ (cont @ Tr)) = Tr)))).
% 3.10/1.61 thf(fact_40_corec_I1_J, axiom, (! [A:$tType,Rt:(A > n),Ct:(A > (set @ ((sum_sum @ t) @ ((sum_sum @ dtree) @ A)))),B3:A]: (((root @ ((((corec @ A) @ Rt) @ Ct) @ B3)) = (Rt @ B3))))).
% 3.10/1.61 thf(fact_41_ssubst__Pair__rhs, axiom, (! [B:$tType,A:$tType,R:A,S:B,R3:(set @ ((product_prod @ A) @ B)),S2:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R) @ S)) @ R3) => ((S2 = S) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R) @ S2)) @ R3)))))).
% 3.10/1.61 thf(fact_42_curry__conv, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((product_curry @ B) @ C) @ A) = (^ [F3:(((product_prod @ B) @ C) > A),A2:B,B2:C]: ((F3 @ ((((product_Pair @ B) @ C) @ A2) @ B2)))))))).
% 3.10/1.61 thf(fact_43_finite__par__c, axiom, (! [Tr12:dtree,Tr22:dtree]: (((finite_finite2 @ ((sum_sum @ t) @ ((product_prod @ dtree) @ dtree))) @ (parall1914194347_par_c @ ((((product_Pair @ dtree) @ dtree) @ Tr12) @ Tr22)))))).
% 3.10/1.61 thf(fact_44_curryI, axiom, (! [A:$tType,B:$tType,F4:(((product_prod @ A) @ B) > $o),A3:A,B3:B]: (((F4 @ ((((product_Pair @ A) @ B) @ A3) @ B3)) => ((((((product_curry @ A) @ B) @ $o) @ F4) @ A3) @ B3))))).
% 3.10/1.61 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A3:A,P:(A > $o)]: (((((member @ A) @ A3) @ ((collect @ A) @ P)) = (P @ A3))))).
% 3.10/1.61 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A6:(set @ A)]: ((((collect @ A) @ (^ [X4:A]: ((((member @ A) @ X4) @ A6)))) = A6)))).
% 3.10/1.61 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 3.10/1.61 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F4:(A > B),G3:(A > B)]: (((! [X3:A]: (((F4 @ X3) = (G3 @ X3)))) => (F4 = G3))))).
% 3.10/1.61 thf(fact_49_swap__simp, axiom, (! [A:$tType,B:$tType,X:B,Y:A]: (((((product_swap @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y)) = ((((product_Pair @ A) @ B) @ Y) @ X))))).
% 3.10/1.61 thf(fact_50_internal__case__prod__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((produc2004651681e_prod @ A) @ B) @ C) = (((product_case_prod @ A) @ B) @ C))))).
% 3.10/1.61 thf(fact_51_swap__swap, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B)]: (((((product_swap @ B) @ A) @ (((product_swap @ A) @ B) @ P2)) = P2)))).
% 3.10/1.61 thf(fact_52_curry__case__prod, axiom, (! [C:$tType,B:$tType,A:$tType,F4:(A > (B > C))]: ((((((product_curry @ A) @ B) @ C) @ ((((product_case_prod @ A) @ B) @ C) @ F4)) = F4)))).
% 3.10/1.61 thf(fact_53_case__prod__curry, axiom, (! [C:$tType,B:$tType,A:$tType,F4:(((product_prod @ A) @ B) > C)]: ((((((product_case_prod @ A) @ B) @ C) @ ((((product_curry @ A) @ B) @ C) @ F4)) = F4)))).
% 3.10/1.61 thf(fact_54_mem__case__prodE, axiom, (! [B:$tType,A:$tType,C:$tType,Z2:A,C2:(B > (C > (set @ A))),P2:((product_prod @ B) @ C)]: (((((member @ A) @ Z2) @ (((((product_case_prod @ B) @ C) @ (set @ A)) @ C2) @ P2)) => (~ ((! [X3:B,Y3:C]: (((P2 = ((((product_Pair @ B) @ C) @ X3) @ Y3)) => (~ ((((member @ A) @ Z2) @ ((C2 @ X3) @ Y3))))))))))))).
% 3.10/1.61 thf(fact_55_old_Oprod_Ocase, axiom, (! [A:$tType,C:$tType,B:$tType,F4:(A > (B > C)),X1:A,X2:B]: (((((((product_case_prod @ A) @ B) @ C) @ F4) @ ((((product_Pair @ A) @ B) @ X1) @ X2)) = ((F4 @ X1) @ X2))))).
% 3.10/1.61 thf(fact_56_dtree__cong, axiom, (! [Tr:dtree,Tr3:dtree]: ((((root @ Tr) = (root @ Tr3)) => (((cont @ Tr) = (cont @ Tr3)) => (Tr = Tr3)))))).
% 3.10/1.61 thf(fact_57_curryE, axiom, (! [A:$tType,B:$tType,F4:(((product_prod @ A) @ B) > $o),A3:A,B3:B]: ((((((((product_curry @ A) @ B) @ $o) @ F4) @ A3) @ B3) => (F4 @ ((((product_Pair @ A) @ B) @ A3) @ B3)))))).
% 3.10/1.61 thf(fact_58_curryD, axiom, (! [A:$tType,B:$tType,F4:(((product_prod @ A) @ B) > $o),A3:A,B3:B]: ((((((((product_curry @ A) @ B) @ $o) @ F4) @ A3) @ B3) => (F4 @ ((((product_Pair @ A) @ B) @ A3) @ B3)))))).
% 3.10/1.61 thf(fact_59_finite__code, axiom, (! [A:$tType]: ((((finite_finite @ A) @ (type2 @ A)) => ((finite_finite2 @ A) = (^ [A7:(set @ A)]: ($true))))))).
% 3.10/1.61 thf(fact_60_split__cong, axiom, (! [C:$tType,B:$tType,A:$tType,Q2:((product_prod @ A) @ B),F4:(A > (B > C)),G3:(A > (B > C)),P2:((product_prod @ A) @ B)]: (((! [X3:A,Y3:B]: (((((((product_Pair @ A) @ B) @ X3) @ Y3) = Q2) => (((F4 @ X3) @ Y3) = ((G3 @ X3) @ Y3))))) => ((P2 = Q2) => ((((((product_case_prod @ A) @ B) @ C) @ F4) @ P2) = (((((product_case_prod @ A) @ B) @ C) @ G3) @ Q2))))))).
% 3.10/1.61 thf(fact_61_same__fstI, axiom, (! [B:$tType,A:$tType,P:(A > $o),X:A,Y6:B,Y:B,R3:(A > (set @ ((product_prod @ B) @ B)))]: (((P @ X) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ Y6) @ Y)) @ (R3 @ X)) => (((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y6)) @ ((((product_Pair @ A) @ B) @ X) @ Y))) @ ((((same_fst @ A) @ B) @ P) @ R3))))))).
% 3.10/1.61 thf(fact_62_cont__Node, axiom, (! [As:(set @ ((sum_sum @ t) @ dtree)),N:n]: ((((finite_finite2 @ ((sum_sum @ t) @ dtree)) @ As) => ((cont @ ((node @ N) @ As)) = As))))).
% 3.10/1.61 thf(fact_63_finite__set__choice, axiom, (! [B:$tType,A:$tType,A6:(set @ A),P:(A > (B > $o))]: ((((finite_finite2 @ A) @ A6) => ((! [X3:A]: (((((member @ A) @ X3) @ A6) => (? [X12:B]: (((P @ X3) @ X12)))))) => (? [F2:(A > B)]: ((! [X5:A]: (((((member @ A) @ X5) @ A6) => ((P @ X5) @ (F2 @ X5)))))))))))).
% 3.10/1.61 thf(fact_64_finite, axiom, (! [A:$tType]: ((((finite_finite @ A) @ (type2 @ A)) => (! [A6:(set @ A)]: (((finite_finite2 @ A) @ A6))))))).
% 3.10/1.61 thf(fact_65_scomp__apply, axiom, (! [A:$tType,D:$tType,C:$tType,B:$tType]: ((((((product_scomp @ B) @ C) @ D) @ A) = (^ [F3:(B > ((product_prod @ C) @ D)),G4:(C > (D > A)),X4:B]: ((((((product_case_prod @ C) @ D) @ A) @ G4) @ (F3 @ X4)))))))).
% 3.10/1.61 thf(fact_66_pair__in__swap__image, axiom, (! [A:$tType,B:$tType,Y:A,X:B,A6:(set @ ((product_prod @ B) @ A))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Y) @ X)) @ ((((image @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ A6)) = (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X) @ Y)) @ A6))))).
% 3.10/1.61 thf(fact_67_finite__imageI, axiom, (! [B:$tType,A:$tType,F5:(set @ A),H:(A > B)]: ((((finite_finite2 @ A) @ F5) => ((finite_finite2 @ B) @ ((((image @ A) @ B) @ H) @ F5)))))).
% 3.10/1.61 thf(fact_68_dtree__simps, axiom, (! [As:(set @ ((sum_sum @ t) @ dtree)),As2:(set @ ((sum_sum @ t) @ dtree)),N:n,N2:n]: ((((finite_finite2 @ ((sum_sum @ t) @ dtree)) @ As) => (((finite_finite2 @ ((sum_sum @ t) @ dtree)) @ As2) => ((((node @ N) @ As) = ((node @ N2) @ As2)) = ((N = N2) & (As = As2)))))))).
% 3.10/1.61 thf(fact_69_finite__cont, axiom, (! [Tr:dtree]: (((finite_finite2 @ ((sum_sum @ t) @ dtree)) @ (cont @ Tr))))).
% 3.10/1.61 thf(fact_70_dtree__cases, axiom, (! [Tr:dtree]: ((~ ((! [N3:n,As3:(set @ ((sum_sum @ t) @ dtree))]: ((((finite_finite2 @ ((sum_sum @ t) @ dtree)) @ As3) => (Tr != ((node @ N3) @ As3)))))))))).
% 3.10/1.61 thf(fact_71_scomp__Pair, axiom, (! [C:$tType,B:$tType,A:$tType,X:(A > ((product_prod @ B) @ C))]: ((((((((product_scomp @ A) @ B) @ C) @ ((product_prod @ B) @ C)) @ X) @ ((product_Pair @ B) @ C)) = X)))).
% 3.10/1.61 thf(fact_72_Pair__scomp, axiom, (! [A:$tType,B:$tType,C:$tType,X:C,F4:(C > (A > B))]: ((((((((product_scomp @ A) @ C) @ A) @ B) @ (((product_Pair @ C) @ A) @ X)) @ F4) = (F4 @ X))))).
% 3.10/1.61 thf(fact_73_scomp__def, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType]: ((((((product_scomp @ A) @ B) @ C) @ D) = (^ [F3:(A > ((product_prod @ B) @ C)),G4:(B > (C > D)),X4:A]: ((((((product_case_prod @ B) @ C) @ D) @ G4) @ (F3 @ X4)))))))).
% 3.10/1.61 thf(fact_74_image__eqI, axiom, (! [A:$tType,B:$tType,B3:A,F4:(B > A),X:B,A6:(set @ B)]: (((B3 = (F4 @ X)) => ((((member @ B) @ X) @ A6) => (((member @ A) @ B3) @ ((((image @ B) @ A) @ F4) @ A6))))))).
% 3.10/1.61 thf(fact_75_surj__swap, axiom, (! [B:$tType,A:$tType]: ((((((image @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ (top_top @ (set @ ((product_prod @ B) @ A)))) = (top_top @ (set @ ((product_prod @ A) @ B))))))).
% 3.10/1.61 thf(fact_76_Sup_OSUP__cong, axiom, (! [A:$tType,B:$tType,A6:(set @ B),B6:(set @ B),C4:(B > A),D3:(B > A),Sup:((set @ A) > A)]: (((A6 = B6) => ((! [X3:B]: (((((member @ B) @ X3) @ B6) => ((C4 @ X3) = (D3 @ X3))))) => ((Sup @ ((((image @ B) @ A) @ C4) @ A6)) = (Sup @ ((((image @ B) @ A) @ D3) @ B6)))))))).
% 3.10/1.61 thf(fact_77_Inf_OINF__cong, axiom, (! [A:$tType,B:$tType,A6:(set @ B),B6:(set @ B),C4:(B > A),D3:(B > A),Inf:((set @ A) > A)]: (((A6 = B6) => ((! [X3:B]: (((((member @ B) @ X3) @ B6) => ((C4 @ X3) = (D3 @ X3))))) => ((Inf @ ((((image @ B) @ A) @ C4) @ A6)) = (Inf @ ((((image @ B) @ A) @ D3) @ B6)))))))).
% 3.10/1.61 thf(fact_78_rev__image__eqI, axiom, (! [B:$tType,A:$tType,X:A,A6:(set @ A),B3:B,F4:(A > B)]: (((((member @ A) @ X) @ A6) => ((B3 = (F4 @ X)) => (((member @ B) @ B3) @ ((((image @ A) @ B) @ F4) @ A6))))))).
% 3.10/1.61 thf(fact_79_ball__imageD, axiom, (! [A:$tType,B:$tType,F4:(B > A),A6:(set @ B),P:(A > $o)]: (((! [X3:A]: (((((member @ A) @ X3) @ ((((image @ B) @ A) @ F4) @ A6)) => (P @ X3)))) => (! [X5:B]: (((((member @ B) @ X5) @ A6) => (P @ (F4 @ X5))))))))).
% 3.10/1.61 thf(fact_80_image__cong, axiom, (! [B:$tType,A:$tType,M:(set @ A),N4:(set @ A),F4:(A > B),G3:(A > B)]: (((M = N4) => ((! [X3:A]: (((((member @ A) @ X3) @ N4) => ((F4 @ X3) = (G3 @ X3))))) => (((((image @ A) @ B) @ F4) @ M) = ((((image @ A) @ B) @ G3) @ N4))))))).
% 3.10/1.61 thf(fact_81_UNIV__I, axiom, (! [A:$tType,X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A)))))).
% 3.10/1.61 thf(fact_82_finite__Plus__UNIV__iff, axiom, (! [A:$tType,B:$tType]: ((((finite_finite2 @ ((sum_sum @ A) @ B)) @ (top_top @ (set @ ((sum_sum @ A) @ B)))) = (((finite_finite2 @ A) @ (top_top @ (set @ A))) & ((finite_finite2 @ B) @ (top_top @ (set @ B)))))))).
% 3.10/1.61 thf(fact_83_Finite__Set_Ofinite__set, axiom, (! [A:$tType]: ((((finite_finite2 @ (set @ A)) @ (top_top @ (set @ (set @ A)))) = ((finite_finite2 @ A) @ (top_top @ (set @ A))))))).
% 3.10/1.61 thf(fact_84_finite__prod, axiom, (! [A:$tType,B:$tType]: ((((finite_finite2 @ ((product_prod @ A) @ B)) @ (top_top @ (set @ ((product_prod @ A) @ B)))) = (((finite_finite2 @ A) @ (top_top @ (set @ A))) & ((finite_finite2 @ B) @ (top_top @ (set @ B)))))))).
% 3.10/1.61 thf(fact_85_finite__Prod__UNIV, axiom, (! [B:$tType,A:$tType]: ((((finite_finite2 @ A) @ (top_top @ (set @ A))) => (((finite_finite2 @ B) @ (top_top @ (set @ B))) => ((finite_finite2 @ ((product_prod @ A) @ B)) @ (top_top @ (set @ ((product_prod @ A) @ B))))))))).
% 3.10/1.61 thf(fact_86_finite__fun__UNIVD2, axiom, (! [A:$tType,B:$tType]: ((((finite_finite2 @ (A > B)) @ (top_top @ (set @ (A > B)))) => ((finite_finite2 @ B) @ (top_top @ (set @ B))))))).
% 3.10/1.61 thf(fact_87_range__eqI, axiom, (! [A:$tType,B:$tType,B3:A,F4:(B > A),X:B]: (((B3 = (F4 @ X)) => (((member @ A) @ B3) @ ((((image @ B) @ A) @ F4) @ (top_top @ (set @ B)))))))).
% 3.10/1.61 thf(fact_88_rangeI, axiom, (! [A:$tType,B:$tType,F4:(B > A),X:B]: ((((member @ A) @ (F4 @ X)) @ ((((image @ B) @ A) @ F4) @ (top_top @ (set @ B))))))).
% 3.10/1.61 thf(fact_89_UNIV__eq__I, axiom, (! [A:$tType,A6:(set @ A)]: (((! [X3:A]: ((((member @ A) @ X3) @ A6))) => ((top_top @ (set @ A)) = A6))))).
% 3.10/1.61 thf(fact_90_UNIV__witness, axiom, (! [A:$tType]: ((? [X3:A]: ((((member @ A) @ X3) @ (top_top @ (set @ A)))))))).
% 3.10/1.61 thf(fact_91_finite__UNIV, axiom, (! [A:$tType]: ((((finite_finite @ A) @ (type2 @ A)) => ((finite_finite2 @ A) @ (top_top @ (set @ A))))))).
% 3.10/1.61 thf(fact_92_infinite__UNIV__char__0, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) @ (type2 @ A)) => (~ (((finite_finite2 @ A) @ (top_top @ (set @ A))))))))).
% 3.10/1.61 thf(fact_93_ex__new__if__finite, axiom, (! [A:$tType,A6:(set @ A)]: (((~ (((finite_finite2 @ A) @ (top_top @ (set @ A))))) => (((finite_finite2 @ A) @ A6) => (? [A5:A]: ((~ ((((member @ A) @ A5) @ A6)))))))))).
% 3.10/1.61 thf(fact_94_imageI, axiom, (! [B:$tType,A:$tType,X:A,A6:(set @ A),F4:(A > B)]: (((((member @ A) @ X) @ A6) => (((member @ B) @ (F4 @ X)) @ ((((image @ A) @ B) @ F4) @ A6)))))).
% 3.10/1.61 thf(fact_95_image__iff, axiom, (! [A:$tType,B:$tType,Z2:A,F4:(B > A),A6:(set @ B)]: (((((member @ A) @ Z2) @ ((((image @ B) @ A) @ F4) @ A6)) = (? [X4:B]: (((((member @ B) @ X4) @ A6) & (Z2 = (F4 @ X4))))))))).
% 3.10/1.61 thf(fact_96_bex__imageD, axiom, (! [A:$tType,B:$tType,F4:(B > A),A6:(set @ B),P:(A > $o)]: (((? [X5:A]: (((((member @ A) @ X5) @ ((((image @ B) @ A) @ F4) @ A6)) & (P @ X5)))) => (? [X3:B]: (((((member @ B) @ X3) @ A6) & (P @ (F4 @ X3))))))))).
% 3.10/1.61 thf(fact_97_finite__option__UNIV, axiom, (! [A:$tType]: ((((finite_finite2 @ (option @ A)) @ (top_top @ (set @ (option @ A)))) = ((finite_finite2 @ A) @ (top_top @ (set @ A))))))).
% 3.10/1.61 thf(fact_98_iso__tuple__UNIV__I, axiom, (! [A:$tType,X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A)))))).
% 3.10/1.61 thf(fact_99_top__apply, axiom, (! [C:$tType,D:$tType]: ((((top @ C) @ (type2 @ C)) => ((top_top @ (D > C)) = (^ [X4:D]: ((top_top @ C)))))))).
% 3.10/1.61 thf(fact_100_surj__def, axiom, (! [B:$tType,A:$tType,F4:(B > A)]: (((((((image @ B) @ A) @ F4) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) = (! [Y5:A]: ((? [X4:B]: ((Y5 = (F4 @ X4)))))))))).
% 3.10/1.61 thf(fact_101_surjI, axiom, (! [B:$tType,A:$tType,G3:(B > A),F4:(A > B)]: (((! [X3:A]: (((G3 @ (F4 @ X3)) = X3))) => (((((image @ B) @ A) @ G3) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 3.10/1.61 thf(fact_102_surjE, axiom, (! [A:$tType,B:$tType,F4:(B > A),Y:A]: (((((((image @ B) @ A) @ F4) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (~ ((! [X3:B]: ((Y != (F4 @ X3)))))))))).
% 3.10/1.61 thf(fact_103_top__set__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (top_top @ (A > $o))))))).
% 3.10/1.61 thf(fact_104_surjD, axiom, (! [A:$tType,B:$tType,F4:(B > A),Y:A]: (((((((image @ B) @ A) @ F4) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (? [X3:B]: ((Y = (F4 @ X3)))))))).
% 3.10/1.61 thf(fact_105_map__prod__surj, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,F4:(A > B),G3:(C > D)]: (((((((image @ A) @ B) @ F4) @ (top_top @ (set @ A))) = (top_top @ (set @ B))) => ((((((image @ C) @ D) @ G3) @ (top_top @ (set @ C))) = (top_top @ (set @ D))) => (((((image @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ D)) @ ((((((product_map_prod @ A) @ B) @ C) @ D) @ F4) @ G3)) @ (top_top @ (set @ ((product_prod @ A) @ C)))) = (top_top @ (set @ ((product_prod @ B) @ D))))))))).
% 3.10/1.61 thf(fact_106_Pow__UNIV, axiom, (! [A:$tType]: ((((pow @ A) @ (top_top @ (set @ A))) = (top_top @ (set @ (set @ A))))))).
% 3.10/1.61 thf(fact_107_surj__swap__iff, axiom, (! [B:$tType,A:$tType,A3:B,B3:B,F4:(B > A)]: (((((((image @ B) @ A) @ (((((swap @ B) @ A) @ A3) @ B3) @ F4)) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) = (((((image @ B) @ A) @ F4) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 3.10/1.61 thf(fact_108_top__empty__eq, axiom, (! [A:$tType]: (((top_top @ (A > $o)) = (^ [X4:A]: ((((member @ A) @ X4) @ (top_top @ (set @ A))))))))).
% 3.10/1.61 thf(fact_109_finite__Pow__iff, axiom, (! [A:$tType,A6:(set @ A)]: ((((finite_finite2 @ (set @ A)) @ ((pow @ A) @ A6)) = ((finite_finite2 @ A) @ A6))))).
% 3.10/1.61 thf(fact_110_top1I, axiom, (! [A:$tType,X:A]: (((top_top @ (A > $o)) @ X)))).
% 3.10/1.61 thf(fact_111_map__prod__simp, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,F4:(C > A),G3:(D > B),A3:C,B3:D]: (((((((((product_map_prod @ C) @ A) @ D) @ B) @ F4) @ G3) @ ((((product_Pair @ C) @ D) @ A3) @ B3)) = ((((product_Pair @ A) @ B) @ (F4 @ A3)) @ (G3 @ B3)))))).
% 3.10/1.61 thf(fact_112_swap__image__eq, axiom, (! [B:$tType,A:$tType,A3:A,A6:(set @ A),B3:A,F4:(A > B)]: (((((member @ A) @ A3) @ A6) => ((((member @ A) @ B3) @ A6) => (((((image @ A) @ B) @ (((((swap @ A) @ B) @ A3) @ B3) @ F4)) @ A6) = ((((image @ A) @ B) @ F4) @ A6))))))).
% 3.10/1.61 thf(fact_113_map__prod__imageI, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,A3:A,B3:B,R3:(set @ ((product_prod @ A) @ B)),F4:(A > C),G3:(B > D)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ R3) => (((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ (F4 @ A3)) @ (G3 @ B3))) @ ((((image @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)) @ ((((((product_map_prod @ A) @ C) @ B) @ D) @ F4) @ G3)) @ R3)))))).
% 3.10/1.61 thf(fact_114_Cantors__paradox, axiom, (! [A:$tType,A6:(set @ A)]: ((~ ((? [X5:(A > (set @ A))]: ((((((image @ A) @ (set @ A)) @ X5) @ A6) = ((pow @ A) @ A6))))))))).
% 3.10/1.61 thf(fact_115_Pow__top, axiom, (! [A:$tType,A6:(set @ A)]: ((((member @ (set @ A)) @ A6) @ ((pow @ A) @ A6))))).
% 3.10/1.61 thf(fact_116_image__Pow__surj, axiom, (! [B:$tType,A:$tType,F4:(B > A),A6:(set @ B),B6:(set @ A)]: (((((((image @ B) @ A) @ F4) @ A6) = B6) => (((((image @ (set @ B)) @ (set @ A)) @ (((image @ B) @ A) @ F4)) @ ((pow @ B) @ A6)) = ((pow @ A) @ B6)))))).
% 3.10/1.61 thf(fact_117_prod__fun__imageE, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,C2:((product_prod @ A) @ B),F4:(C > A),G3:(D > B),R3:(set @ ((product_prod @ C) @ D))]: (((((member @ ((product_prod @ A) @ B)) @ C2) @ ((((image @ ((product_prod @ C) @ D)) @ ((product_prod @ A) @ B)) @ ((((((product_map_prod @ C) @ A) @ D) @ B) @ F4) @ G3)) @ R3)) => (~ ((! [X3:C,Y3:D]: (((C2 = ((((product_Pair @ A) @ B) @ (F4 @ X3)) @ (G3 @ Y3))) => (~ ((((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ X3) @ Y3)) @ R3)))))))))))).
% 3.10/1.61 thf(fact_118_surj__imp__surj__swap, axiom, (! [B:$tType,A:$tType,F4:(B > A),A3:B,B3:B]: (((((((image @ B) @ A) @ F4) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (((((image @ B) @ A) @ (((((swap @ B) @ A) @ A3) @ B3) @ F4)) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 3.10/1.61 thf(fact_119_top__conj_I2_J, axiom, (! [A:$tType,P:$o,X:A]: (((P & ((top_top @ (A > $o)) @ X)) = P)))).
% 3.10/1.61 thf(fact_120_top__conj_I1_J, axiom, (! [A:$tType,X:A,P:$o]: (((((top_top @ (A > $o)) @ X) & P) = P)))).
% 3.10/1.61 thf(fact_121_range__fst, axiom, (! [B:$tType,A:$tType]: ((((((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ (top_top @ (set @ ((product_prod @ A) @ B)))) = (top_top @ (set @ A)))))).
% 3.10/1.61 thf(fact_122_range__snd, axiom, (! [B:$tType,A:$tType]: ((((((image @ ((product_prod @ B) @ A)) @ A) @ ((product_snd @ B) @ A)) @ (top_top @ (set @ ((product_prod @ B) @ A)))) = (top_top @ (set @ A)))))).
% 3.10/1.61 thf(fact_123_in__inv__image, axiom, (! [A:$tType,B:$tType,X:A,Y:A,R:(set @ ((product_prod @ B) @ B)),F4:(A > B)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((((inv_image @ B) @ A) @ R) @ F4)) = (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ (F4 @ X)) @ (F4 @ Y))) @ R))))).
% 3.10/1.61 thf(fact_124_fst__map__prod, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,F4:(C > A),G3:(D > B),X:((product_prod @ C) @ D)]: (((((product_fst @ A) @ B) @ (((((((product_map_prod @ C) @ A) @ D) @ B) @ F4) @ G3) @ X)) = (F4 @ (((product_fst @ C) @ D) @ X)))))).
% 3.10/1.61 thf(fact_125_snd__map__prod, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,F4:(C > B),G3:(D > A),X:((product_prod @ C) @ D)]: (((((product_snd @ B) @ A) @ (((((((product_map_prod @ C) @ B) @ D) @ A) @ F4) @ G3) @ X)) = (G3 @ (((product_snd @ C) @ D) @ X)))))).
% 3.10/1.61 thf(fact_126_prod_Ocollapse, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)) = Prod)))).
% 3.10/1.61 thf(fact_127_snd__swap, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B)]: (((((product_snd @ B) @ A) @ (((product_swap @ A) @ B) @ X)) = (((product_fst @ A) @ B) @ X))))).
% 3.10/1.61 thf(fact_128_fst__swap, axiom, (! [A:$tType,B:$tType,X:((product_prod @ B) @ A)]: (((((product_fst @ A) @ B) @ (((product_swap @ B) @ A) @ X)) = (((product_snd @ B) @ A) @ X))))).
% 3.10/1.61 thf(fact_129_prod_Osplit__sel, axiom, (! [C:$tType,B:$tType,A:$tType,P:(C > $o),F4:(A > (B > C)),Prod:((product_prod @ A) @ B)]: (((P @ (((((product_case_prod @ A) @ B) @ C) @ F4) @ Prod)) = ((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod))) => (P @ ((F4 @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))))).
% 3.10/1.61 thf(fact_130_prod_Osplit__sel__asm, axiom, (! [C:$tType,B:$tType,A:$tType,P:(C > $o),F4:(A > (B > C)),Prod:((product_prod @ A) @ B)]: (((P @ (((((product_case_prod @ A) @ B) @ C) @ F4) @ Prod)) = (~ (((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod))) & (~ ((P @ ((F4 @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))))))))).
% 3.10/1.61 thf(fact_131_prod_Oexhaust__sel, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))).
% 3.10/1.61 thf(fact_132_surjective__pairing, axiom, (! [B:$tType,A:$tType,T2:((product_prod @ A) @ B)]: ((T2 = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ T2)) @ (((product_snd @ A) @ B) @ T2)))))).
% 3.10/1.61 thf(fact_133_prod__eqI, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B),Q2:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ P2) = (((product_fst @ A) @ B) @ Q2)) => (((((product_snd @ A) @ B) @ P2) = (((product_snd @ A) @ B) @ Q2)) => (P2 = Q2)))))).
% 3.10/1.61 thf(fact_134_prod_Oexpand, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B),Prod2:((product_prod @ A) @ B)]: (((((((product_fst @ A) @ B) @ Prod) = (((product_fst @ A) @ B) @ Prod2)) & ((((product_snd @ A) @ B) @ Prod) = (((product_snd @ A) @ B) @ Prod2))) => (Prod = Prod2))))).
% 3.10/1.61 thf(fact_135_prod__eq__iff, axiom, (! [B:$tType,A:$tType]: (((^ [Y7:((product_prod @ A) @ B),Z3:((product_prod @ A) @ B)]: ((Y7 = Z3))) = (^ [S3:((product_prod @ A) @ B),T3:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ S3) = (((product_fst @ A) @ B) @ T3)) & ((((product_snd @ A) @ B) @ S3) = (((product_snd @ A) @ B) @ T3))))))))).
% 3.10/1.61 thf(fact_136_case__prod__beta, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((product_case_prod @ B) @ C) @ A) = (^ [F3:(B > (C > A)),P3:((product_prod @ B) @ C)]: (((F3 @ (((product_fst @ B) @ C) @ P3)) @ (((product_snd @ B) @ C) @ P3)))))))).
% 3.10/1.61 thf(fact_137_prod_Ocase__eq__if, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((product_case_prod @ A) @ B) @ C) = (^ [F3:(A > (B > C)),Prod3:((product_prod @ A) @ B)]: (((F3 @ (((product_fst @ A) @ B) @ Prod3)) @ (((product_snd @ A) @ B) @ Prod3)))))))).
% 3.10/1.61 thf(fact_138_Product__Type_OCollect__case__prodD, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B),A6:(A > (B > $o))]: (((((member @ ((product_prod @ A) @ B)) @ X) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ A6))) => ((A6 @ (((product_fst @ A) @ B) @ X)) @ (((product_snd @ A) @ B) @ X)))))).
% 3.10/1.61 thf(fact_139_fst__conv, axiom, (! [B:$tType,A:$tType,X1:A,X2:B]: (((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X1) @ X2)) = X1)))).
% 3.10/1.61 thf(fact_140_fst__eqD, axiom, (! [B:$tType,A:$tType,X:A,Y:B,A3:A]: ((((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X) @ Y)) = A3) => (X = A3))))).
% 3.10/1.61 thf(fact_141_snd__conv, axiom, (! [Aa:$tType,A:$tType,X1:Aa,X2:A]: (((((product_snd @ Aa) @ A) @ ((((product_Pair @ Aa) @ A) @ X1) @ X2)) = X2)))).
% 3.10/1.61 thf(fact_142_snd__eqD, axiom, (! [B:$tType,A:$tType,X:B,Y:A,A3:A]: ((((((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y)) = A3) => (Y = A3))))).
% 3.10/1.61 thf(fact_143_prod_Oswap__def, axiom, (! [B:$tType,A:$tType]: ((((product_swap @ A) @ B) = (^ [P3:((product_prod @ A) @ B)]: (((((product_Pair @ B) @ A) @ (((product_snd @ A) @ B) @ P3)) @ (((product_fst @ A) @ B) @ P3)))))))).
% 3.10/1.61 thf(fact_144_exI__realizer, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),Y:A,X:B]: ((((P @ Y) @ X) => ((P @ (((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y))) @ (((product_fst @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y))))))).
% 3.10/1.61 thf(fact_145_conjI__realizer, axiom, (! [A:$tType,B:$tType,P:(A > $o),P2:A,Q:(B > $o),Q2:B]: (((P @ P2) => ((Q @ Q2) => ((P @ (((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ P2) @ Q2))) & (Q @ (((product_snd @ A) @ B) @ ((((product_Pair @ A) @ B) @ P2) @ Q2))))))))).
% 3.10/1.61 thf(fact_146_BNF__Greatest__Fixpoint_Osubst__Pair, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),X:A,Y:B,A3:((product_prod @ A) @ B)]: ((((P @ X) @ Y) => ((A3 = ((((product_Pair @ A) @ B) @ X) @ Y)) => ((P @ (((product_fst @ A) @ B) @ A3)) @ (((product_snd @ A) @ B) @ A3))))))).
% 3.10/1.61 thf(fact_147_sndI, axiom, (! [A:$tType,B:$tType,X:((product_prod @ A) @ B),Y:A,Z2:B]: (((X = ((((product_Pair @ A) @ B) @ Y) @ Z2)) => ((((product_snd @ A) @ B) @ X) = Z2))))).
% 3.10/1.61 thf(fact_148_fstI, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B),Y:A,Z2:B]: (((X = ((((product_Pair @ A) @ B) @ Y) @ Z2)) => ((((product_fst @ A) @ B) @ X) = Y))))).
% 3.10/1.61 thf(fact_149_eq__snd__iff, axiom, (! [A:$tType,B:$tType,B3:A,P2:((product_prod @ B) @ A)]: (((B3 = (((product_snd @ B) @ A) @ P2)) = (? [A2:B]: ((P2 = ((((product_Pair @ B) @ A) @ A2) @ B3)))))))).
% 3.10/1.61 thf(fact_150_eq__fst__iff, axiom, (! [A:$tType,B:$tType,A3:A,P2:((product_prod @ A) @ B)]: (((A3 = (((product_fst @ A) @ B) @ P2)) = (? [B2:B]: ((P2 = ((((product_Pair @ A) @ B) @ A3) @ B2)))))))).
% 3.10/1.61 thf(fact_151_Collect__split__mono__strong, axiom, (! [B:$tType,A:$tType,X6:(set @ A),A6:(set @ ((product_prod @ A) @ B)),Y8:(set @ B),P:(A > (B > $o)),Q:(A > (B > $o))]: (((X6 = ((((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ A6)) => ((Y8 = ((((image @ ((product_prod @ A) @ B)) @ B) @ ((product_snd @ A) @ B)) @ A6)) => ((! [X3:A]: (((((member @ A) @ X3) @ X6) => (! [Xa:B]: (((((member @ B) @ Xa) @ Y8) => (((P @ X3) @ Xa) => ((Q @ X3) @ Xa)))))))) => ((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ A6) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ P))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ A6) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ Q)))))))))).
% 3.10/1.61 thf(fact_152_subsetI, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A)]: (((! [X3:A]: (((((member @ A) @ X3) @ A6) => (((member @ A) @ X3) @ B6)))) => (((ord_less_eq @ (set @ A)) @ A6) @ B6))))).
% 3.10/1.61 thf(fact_153_subset__antisym, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) => ((((ord_less_eq @ (set @ A)) @ B6) @ A6) => (A6 = B6)))))).
% 3.10/1.61 thf(fact_154_Pow__iff, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A)]: (((((member @ (set @ A)) @ A6) @ ((pow @ A) @ B6)) = (((ord_less_eq @ (set @ A)) @ A6) @ B6))))).
% 3.10/1.61 thf(fact_155_PowI, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) => (((member @ (set @ A)) @ A6) @ ((pow @ A) @ B6)))))).
% 3.10/1.61 thf(fact_156_subrelI, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ A) @ B))]: (((! [X3:A,Y3:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ R) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ S)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R) @ S))))).
% 3.10/1.61 thf(fact_157_Pow__mono, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) => (((ord_less_eq @ (set @ (set @ A))) @ ((pow @ A) @ A6)) @ ((pow @ A) @ B6)))))).
% 3.10/1.61 thf(fact_158_top_Oextremum__uniqueI, axiom, (! [A:$tType]: ((((order_top @ A) @ (type2 @ A)) => (! [A3:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A3) => (A3 = (top_top @ A))))))))).
% 3.10/1.61 thf(fact_159_top_Oextremum__unique, axiom, (! [A:$tType]: ((((order_top @ A) @ (type2 @ A)) => (! [A3:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A3) = (A3 = (top_top @ A))))))))).
% 3.10/1.61 thf(fact_160_top__greatest, axiom, (! [A:$tType]: ((((order_top @ A) @ (type2 @ A)) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ (top_top @ A)))))))).
% 3.10/1.61 thf(fact_161_rev__finite__subset, axiom, (! [A:$tType,B6:(set @ A),A6:(set @ A)]: ((((finite_finite2 @ A) @ B6) => ((((ord_less_eq @ (set @ A)) @ A6) @ B6) => ((finite_finite2 @ A) @ A6)))))).
% 3.10/1.61 thf(fact_162_infinite__super, axiom, (! [A:$tType,S4:(set @ A),T4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ S4) @ T4) => ((~ (((finite_finite2 @ A) @ S4))) => (~ (((finite_finite2 @ A) @ T4)))))))).
% 3.10/1.61 thf(fact_163_finite__subset, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) => (((finite_finite2 @ A) @ B6) => ((finite_finite2 @ A) @ A6)))))).
% 3.10/1.61 thf(fact_164_subset__UNIV, axiom, (! [A:$tType,A6:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A6) @ (top_top @ (set @ A)))))).
% 3.10/1.61 thf(fact_165_set__mp, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) => ((((member @ A) @ X) @ A6) => (((member @ A) @ X) @ B6)))))).
% 3.10/1.61 thf(fact_166_in__mono, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) => ((((member @ A) @ X) @ A6) => (((member @ A) @ X) @ B6)))))).
% 3.10/1.61 thf(fact_167_subsetD, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) => ((((member @ A) @ C2) @ A6) => (((member @ A) @ C2) @ B6)))))).
% 3.10/1.61 thf(fact_168_subsetCE, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) => ((((member @ A) @ C2) @ A6) => (((member @ A) @ C2) @ B6)))))).
% 3.10/1.61 thf(fact_169_equalityE, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A)]: (((A6 = B6) => (~ (((((ord_less_eq @ (set @ A)) @ A6) @ B6) => (~ ((((ord_less_eq @ (set @ A)) @ B6) @ A6)))))))))).
% 3.10/1.61 thf(fact_170_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A7:(set @ A),B7:(set @ A)]: ((! [X4:A]: (((((member @ A) @ X4) @ A7) => (((member @ A) @ X4) @ B7)))))))))).
% 3.10/1.61 thf(fact_171_equalityD1, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A)]: (((A6 = B6) => (((ord_less_eq @ (set @ A)) @ A6) @ B6))))).
% 3.10/1.61 thf(fact_172_equalityD2, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A)]: (((A6 = B6) => (((ord_less_eq @ (set @ A)) @ B6) @ A6))))).
% 3.10/1.61 thf(fact_173_set__rev__mp, axiom, (! [A:$tType,X:A,A6:(set @ A),B6:(set @ A)]: (((((member @ A) @ X) @ A6) => ((((ord_less_eq @ (set @ A)) @ A6) @ B6) => (((member @ A) @ X) @ B6)))))).
% 3.10/1.61 thf(fact_174_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A7:(set @ A),B7:(set @ A)]: ((! [T3:A]: (((((member @ A) @ T3) @ A7) => (((member @ A) @ T3) @ B7)))))))))).
% 3.10/1.61 thf(fact_175_rev__subsetD, axiom, (! [A:$tType,C2:A,A6:(set @ A),B6:(set @ A)]: (((((member @ A) @ C2) @ A6) => ((((ord_less_eq @ (set @ A)) @ A6) @ B6) => (((member @ A) @ C2) @ B6)))))).
% 3.10/1.61 thf(fact_176_subset__refl, axiom, (! [A:$tType,A6:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A6) @ A6)))).
% 3.10/1.61 thf(fact_177_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) => (Q @ X3)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 3.10/1.61 thf(fact_178_subset__trans, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) => ((((ord_less_eq @ (set @ A)) @ B6) @ C4) => (((ord_less_eq @ (set @ A)) @ A6) @ C4)))))).
% 3.10/1.61 thf(fact_179_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y7:(set @ A),Z3:(set @ A)]: ((Y7 = Z3))) = (^ [A7:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ B7) & (((ord_less_eq @ (set @ A)) @ B7) @ A7)))))))).
% 3.10/1.61 thf(fact_180_contra__subsetD, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) => ((~ ((((member @ A) @ C2) @ B6))) => (~ ((((member @ A) @ C2) @ A6)))))))).
% 3.10/1.61 thf(fact_181_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X4:A]: (((P @ X4) => (Q @ X4)))))))).
% 3.10/1.61 thf(fact_182_PowD, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A)]: (((((member @ (set @ A)) @ A6) @ ((pow @ A) @ B6)) => (((ord_less_eq @ (set @ A)) @ A6) @ B6))))).
% 3.10/1.61 thf(fact_183_image__mono, axiom, (! [B:$tType,A:$tType,A6:(set @ A),B6:(set @ A),F4:(A > B)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F4) @ A6)) @ ((((image @ A) @ B) @ F4) @ B6)))))).
% 3.10/1.61 thf(fact_184_image__subsetI, axiom, (! [A:$tType,B:$tType,A6:(set @ A),F4:(A > B),B6:(set @ B)]: (((! [X3:A]: (((((member @ A) @ X3) @ A6) => (((member @ B) @ (F4 @ X3)) @ B6)))) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F4) @ A6)) @ B6))))).
% 3.10/1.61 thf(fact_185_subset__imageE, axiom, (! [A:$tType,B:$tType,B6:(set @ A),F4:(B > A),A6:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B6) @ ((((image @ B) @ A) @ F4) @ A6)) => (~ ((! [C5:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ C5) @ A6) => (B6 != ((((image @ B) @ A) @ F4) @ C5))))))))))).
% 3.10/1.61 thf(fact_186_image__subset__iff, axiom, (! [A:$tType,B:$tType,F4:(B > A),A6:(set @ B),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F4) @ A6)) @ B6) = (! [X4:B]: (((((member @ B) @ X4) @ A6) => (((member @ A) @ (F4 @ X4)) @ B6)))))))).
% 3.10/1.61 thf(fact_187_subset__image__iff, axiom, (! [A:$tType,B:$tType,B6:(set @ A),F4:(B > A),A6:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B6) @ ((((image @ B) @ A) @ F4) @ A6)) = (? [AA:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ AA) @ A6) & (B6 = ((((image @ B) @ A) @ F4) @ AA))))))))).
% 3.10/1.61 thf(fact_188_image__Pow__mono, axiom, (! [B:$tType,A:$tType,F4:(B > A),A6:(set @ B),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F4) @ A6)) @ B6) => (((ord_less_eq @ (set @ (set @ A))) @ ((((image @ (set @ B)) @ (set @ A)) @ (((image @ B) @ A) @ F4)) @ ((pow @ B) @ A6))) @ ((pow @ A) @ B6)))))).
% 3.10/1.61 thf(fact_189_finite__surj, axiom, (! [A:$tType,B:$tType,A6:(set @ A),B6:(set @ B),F4:(A > B)]: ((((finite_finite2 @ A) @ A6) => ((((ord_less_eq @ (set @ B)) @ B6) @ ((((image @ A) @ B) @ F4) @ A6)) => ((finite_finite2 @ B) @ B6)))))).
% 3.10/1.61 thf(fact_190_finite__subset__image, axiom, (! [A:$tType,B:$tType,B6:(set @ A),F4:(B > A),A6:(set @ B)]: ((((finite_finite2 @ A) @ B6) => ((((ord_less_eq @ (set @ A)) @ B6) @ ((((image @ B) @ A) @ F4) @ A6)) => (? [C5:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ C5) @ A6) & (((finite_finite2 @ B) @ C5) & (B6 = ((((image @ B) @ A) @ F4) @ C5))))))))))).
% 3.10/1.61 thf(fact_191_relChain__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (((bNF_Ca1785829860lChain @ A) @ B) = (^ [R2:(set @ ((product_prod @ A) @ A)),As4:(A > B)]: ((! [I:A,J:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ I) @ J)) @ R2) => (((ord_less_eq @ B) @ (As4 @ I)) @ (As4 @ J)))))))))))).
% 3.10/1.61 thf(fact_192_fstOp__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((bNF_fstOp @ A) @ B) @ C) = (^ [P4:(A > (B > $o)),Q3:(B > (C > $o)),Ac:((product_prod @ A) @ C)]: (((((product_Pair @ A) @ B) @ (((product_fst @ A) @ C) @ Ac)) @ (((((((bNF_pick_middlep @ A) @ B) @ C) @ P4) @ Q3) @ (((product_fst @ A) @ C) @ Ac)) @ (((product_snd @ A) @ C) @ Ac))))))))).
% 3.10/1.61 thf(fact_193_sndOp__def, axiom, (! [B:$tType,A:$tType,C:$tType]: (((((bNF_sndOp @ C) @ A) @ B) = (^ [P4:(C > (A > $o)),Q3:(A > (B > $o)),Ac:((product_prod @ C) @ B)]: (((((product_Pair @ A) @ B) @ (((((((bNF_pick_middlep @ C) @ A) @ B) @ P4) @ Q3) @ (((product_fst @ C) @ B) @ Ac)) @ (((product_snd @ C) @ B) @ Ac))) @ (((product_snd @ C) @ B) @ Ac)))))))).
% 3.10/1.61 thf(fact_194_accp__subset, axiom, (! [A:$tType,R1:(A > (A > $o)),R22:(A > (A > $o))]: (((((ord_less_eq @ (A > (A > $o))) @ R1) @ R22) => (((ord_less_eq @ (A > $o)) @ ((accp @ A) @ R22)) @ ((accp @ A) @ R1)))))).
% 3.10/1.61 thf(fact_195_accp__subset__induct, axiom, (! [A:$tType,D3:(A > $o),R3:(A > (A > $o)),X:A,P:(A > $o)]: (((((ord_less_eq @ (A > $o)) @ D3) @ ((accp @ A) @ R3)) => ((! [X3:A,Z:A]: (((D3 @ X3) => (((R3 @ Z) @ X3) => (D3 @ Z))))) => ((D3 @ X) => ((! [X3:A]: (((D3 @ X3) => ((! [Z4:A]: ((((R3 @ Z4) @ X3) => (P @ Z4)))) => (P @ X3))))) => (P @ X)))))))).
% 3.10/1.61 thf(fact_196_Collect__case__prod__mono, axiom, (! [B:$tType,A:$tType,A6:(A > (B > $o)),B6:(A > (B > $o))]: (((((ord_less_eq @ (A > (B > $o))) @ A6) @ B6) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ A6))) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ B6))))))).
% 3.10/1.61 thf(fact_197_finite__vimageD_H, axiom, (! [A:$tType,B:$tType,F4:(A > B),A6:(set @ B)]: ((((finite_finite2 @ A) @ ((((vimage @ A) @ B) @ F4) @ A6)) => ((((ord_less_eq @ (set @ B)) @ A6) @ ((((image @ A) @ B) @ F4) @ (top_top @ (set @ A)))) => ((finite_finite2 @ B) @ A6)))))).
% 3.10/1.61 thf(fact_198_image2__eqI, axiom, (! [A:$tType,C:$tType,B:$tType,B3:A,F4:(B > A),X:B,C2:C,G3:(B > C),A6:(set @ B)]: (((B3 = (F4 @ X)) => ((C2 = (G3 @ X)) => ((((member @ B) @ X) @ A6) => (((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ B3) @ C2)) @ ((((((bNF_Greatest_image2 @ B) @ A) @ C) @ A6) @ F4) @ G3)))))))).
% 3.10/1.61 thf(fact_199_vimageI, axiom, (! [B:$tType,A:$tType,F4:(B > A),A3:B,B3:A,B6:(set @ A)]: ((((F4 @ A3) = B3) => ((((member @ A) @ B3) @ B6) => (((member @ B) @ A3) @ ((((vimage @ B) @ A) @ F4) @ B6))))))).
% 3.10/1.61 thf(fact_200_vimage__eq, axiom, (! [A:$tType,B:$tType,A3:A,F4:(A > B),B6:(set @ B)]: (((((member @ A) @ A3) @ ((((vimage @ A) @ B) @ F4) @ B6)) = (((member @ B) @ (F4 @ A3)) @ B6))))).
% 3.10/1.61 thf(fact_201_vimage__UNIV, axiom, (! [B:$tType,A:$tType,F4:(A > B)]: ((((((vimage @ A) @ B) @ F4) @ (top_top @ (set @ B))) = (top_top @ (set @ A)))))).
% 3.10/1.61 thf(fact_202_vimage__mono, axiom, (! [B:$tType,A:$tType,A6:(set @ A),B6:(set @ A),F4:(B > A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) => (((ord_less_eq @ (set @ B)) @ ((((vimage @ B) @ A) @ F4) @ A6)) @ ((((vimage @ B) @ A) @ F4) @ B6)))))).
% 3.10/1.61 thf(fact_203_image__subset__iff__subset__vimage, axiom, (! [B:$tType,A:$tType,F4:(B > A),A6:(set @ B),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F4) @ A6)) @ B6) = (((ord_less_eq @ (set @ B)) @ A6) @ ((((vimage @ B) @ A) @ F4) @ B6)))))).
% 3.10/1.61 thf(fact_204_image__vimage__subset, axiom, (! [B:$tType,A:$tType,F4:(B > A),A6:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F4) @ ((((vimage @ B) @ A) @ F4) @ A6))) @ A6)))).
% 3.10/1.61 thf(fact_205_surj__image__vimage__eq, axiom, (! [B:$tType,A:$tType,F4:(B > A),A6:(set @ A)]: (((((((image @ B) @ A) @ F4) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (((((image @ B) @ A) @ F4) @ ((((vimage @ B) @ A) @ F4) @ A6)) = A6))))).
% 3.10/1.61 thf(fact_206_vimageD, axiom, (! [A:$tType,B:$tType,A3:A,F4:(A > B),A6:(set @ B)]: (((((member @ A) @ A3) @ ((((vimage @ A) @ B) @ F4) @ A6)) => (((member @ B) @ (F4 @ A3)) @ A6))))).
% 3.10/1.61 thf(fact_207_vimageE, axiom, (! [A:$tType,B:$tType,A3:A,F4:(A > B),B6:(set @ B)]: (((((member @ A) @ A3) @ ((((vimage @ A) @ B) @ F4) @ B6)) => (((member @ B) @ (F4 @ A3)) @ B6))))).
% 3.10/1.61 thf(fact_208_vimageI2, axiom, (! [B:$tType,A:$tType,F4:(B > A),A3:B,A6:(set @ A)]: (((((member @ A) @ (F4 @ A3)) @ A6) => (((member @ B) @ A3) @ ((((vimage @ B) @ A) @ F4) @ A6)))))).
% 3.10/1.61 thf(fact_209_vimage__Collect, axiom, (! [B:$tType,A:$tType,P:(B > $o),F4:(A > B),Q:(A > $o)]: (((! [X3:A]: (((P @ (F4 @ X3)) = (Q @ X3)))) => (((((vimage @ A) @ B) @ F4) @ ((collect @ B) @ P)) = ((collect @ A) @ Q)))))).
% 3.10/1.61 thf(fact_210_finite__vimageD, axiom, (! [A:$tType,B:$tType,H:(A > B),F5:(set @ B)]: ((((finite_finite2 @ A) @ ((((vimage @ A) @ B) @ H) @ F5)) => ((((((image @ A) @ B) @ H) @ (top_top @ (set @ A))) = (top_top @ (set @ B))) => ((finite_finite2 @ B) @ F5)))))).
% 3.10/1.61 thf(fact_211_vimage__subsetD, axiom, (! [A:$tType,B:$tType,F4:(B > A),B6:(set @ A),A6:(set @ B)]: (((((((image @ B) @ A) @ F4) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => ((((ord_less_eq @ (set @ B)) @ ((((vimage @ B) @ A) @ F4) @ B6)) @ A6) => (((ord_less_eq @ (set @ A)) @ B6) @ ((((image @ B) @ A) @ F4) @ A6))))))).
% 3.10/1.61 thf(fact_212_image__vimage__eq, axiom, (! [A:$tType,B:$tType,F4:(B > A),A6:(set @ A)]: ((((((image @ B) @ A) @ F4) @ ((((vimage @ B) @ A) @ F4) @ A6)) = (((inf_inf @ (set @ A)) @ A6) @ ((((image @ B) @ A) @ F4) @ (top_top @ (set @ B)))))))).
% 3.10/1.61 thf(fact_213_vimage__subsetI, axiom, (! [B:$tType,A:$tType,F4:(A > B),B6:(set @ B),A6:(set @ A)]: ((((((inj_on @ A) @ B) @ F4) @ (top_top @ (set @ A))) => ((((ord_less_eq @ (set @ B)) @ B6) @ ((((image @ A) @ B) @ F4) @ A6)) => (((ord_less_eq @ (set @ A)) @ ((((vimage @ A) @ B) @ F4) @ B6)) @ A6)))))).
% 3.10/1.61 thf(fact_214_Int__iff, axiom, (! [A:$tType,C2:A,A6:(set @ A),B6:(set @ A)]: (((((member @ A) @ C2) @ (((inf_inf @ (set @ A)) @ A6) @ B6)) = ((((member @ A) @ C2) @ A6) & (((member @ A) @ C2) @ B6)))))).
% 3.10/1.61 thf(fact_215_IntI, axiom, (! [A:$tType,C2:A,A6:(set @ A),B6:(set @ A)]: (((((member @ A) @ C2) @ A6) => ((((member @ A) @ C2) @ B6) => (((member @ A) @ C2) @ (((inf_inf @ (set @ A)) @ A6) @ B6))))))).
% 3.10/1.61 thf(fact_216_finite__Int, axiom, (! [A:$tType,F5:(set @ A),G5:(set @ A)]: (((((finite_finite2 @ A) @ F5) | ((finite_finite2 @ A) @ G5)) => ((finite_finite2 @ A) @ (((inf_inf @ (set @ A)) @ F5) @ G5)))))).
% 3.10/1.61 thf(fact_217_Int__subset__iff, axiom, (! [A:$tType,C4:(set @ A),A6:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ C4) @ (((inf_inf @ (set @ A)) @ A6) @ B6)) = ((((ord_less_eq @ (set @ A)) @ C4) @ A6) & (((ord_less_eq @ (set @ A)) @ C4) @ B6)))))).
% 3.10/1.61 thf(fact_218_Int__UNIV, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A)]: ((((((inf_inf @ (set @ A)) @ A6) @ B6) = (top_top @ (set @ A))) = ((A6 = (top_top @ (set @ A))) & (B6 = (top_top @ (set @ A)))))))).
% 3.10/1.61 thf(fact_219_vimage__Int, axiom, (! [A:$tType,B:$tType,F4:(A > B),A6:(set @ B),B6:(set @ B)]: ((((((vimage @ A) @ B) @ F4) @ (((inf_inf @ (set @ B)) @ A6) @ B6)) = (((inf_inf @ (set @ A)) @ ((((vimage @ A) @ B) @ F4) @ A6)) @ ((((vimage @ A) @ B) @ F4) @ B6)))))).
% 3.10/1.61 thf(fact_220_Pow__Int__eq, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A)]: ((((pow @ A) @ (((inf_inf @ (set @ A)) @ A6) @ B6)) = (((inf_inf @ (set @ (set @ A))) @ ((pow @ A) @ A6)) @ ((pow @ A) @ B6)))))).
% 3.10/1.61 thf(fact_221_Int__mono, axiom, (! [A:$tType,A6:(set @ A),C4:(set @ A),B6:(set @ A),D3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ C4) => ((((ord_less_eq @ (set @ A)) @ B6) @ D3) => (((ord_less_eq @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A6) @ B6)) @ (((inf_inf @ (set @ A)) @ C4) @ D3))))))).
% 3.10/1.61 thf(fact_222_Int__lower1, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A6) @ B6)) @ A6)))).
% 3.10/1.61 thf(fact_223_Int__lower2, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A6) @ B6)) @ B6)))).
% 3.10/1.61 thf(fact_224_Int__absorb1, axiom, (! [A:$tType,B6:(set @ A),A6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B6) @ A6) => ((((inf_inf @ (set @ A)) @ A6) @ B6) = B6))))).
% 3.10/1.61 thf(fact_225_Int__absorb2, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) => ((((inf_inf @ (set @ A)) @ A6) @ B6) = A6))))).
% 3.10/1.61 thf(fact_226_Int__greatest, axiom, (! [A:$tType,C4:(set @ A),A6:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ C4) @ A6) => ((((ord_less_eq @ (set @ A)) @ C4) @ B6) => (((ord_less_eq @ (set @ A)) @ C4) @ (((inf_inf @ (set @ A)) @ A6) @ B6))))))).
% 3.10/1.61 thf(fact_227_Int__Collect__mono, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A),P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B6) => ((! [X3:A]: (((((member @ A) @ X3) @ A6) => ((P @ X3) => (Q @ X3))))) => (((ord_less_eq @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A6) @ ((collect @ A) @ P))) @ (((inf_inf @ (set @ A)) @ B6) @ ((collect @ A) @ Q)))))))).
% 3.10/1.61 thf(fact_228_inj__on__image__Int, axiom, (! [B:$tType,A:$tType,F4:(A > B),C4:(set @ A),A6:(set @ A),B6:(set @ A)]: ((((((inj_on @ A) @ B) @ F4) @ C4) => ((((ord_less_eq @ (set @ A)) @ A6) @ C4) => ((((ord_less_eq @ (set @ A)) @ B6) @ C4) => (((((image @ A) @ B) @ F4) @ (((inf_inf @ (set @ A)) @ A6) @ B6)) = (((inf_inf @ (set @ B)) @ ((((image @ A) @ B) @ F4) @ A6)) @ ((((image @ A) @ B) @ F4) @ B6))))))))).
% 3.10/1.61 thf(fact_229_inj__on__image__eq__iff, axiom, (! [B:$tType,A:$tType,F4:(A > B),C4:(set @ A),A6:(set @ A),B6:(set @ A)]: ((((((inj_on @ A) @ B) @ F4) @ C4) => ((((ord_less_eq @ (set @ A)) @ A6) @ C4) => ((((ord_less_eq @ (set @ A)) @ B6) @ C4) => ((((((image @ A) @ B) @ F4) @ A6) = ((((image @ A) @ B) @ F4) @ B6)) = (A6 = B6)))))))).
% 3.10/1.61 thf(fact_230_inj__on__image__mem__iff, axiom, (! [B:$tType,A:$tType,F4:(A > B),B6:(set @ A),A3:A,A6:(set @ A)]: ((((((inj_on @ A) @ B) @ F4) @ B6) => ((((member @ A) @ A3) @ B6) => ((((ord_less_eq @ (set @ A)) @ A6) @ B6) => ((((member @ B) @ (F4 @ A3)) @ ((((image @ A) @ B) @ F4) @ A6)) = (((member @ A) @ A3) @ A6)))))))).
% 3.10/1.61 thf(fact_231_inj__on__image__mem__iff__alt, axiom, (! [B:$tType,A:$tType,F4:(A > B),B6:(set @ A),A6:(set @ A),A3:A]: ((((((inj_on @ A) @ B) @ F4) @ B6) => ((((ord_less_eq @ (set @ A)) @ A6) @ B6) => ((((member @ B) @ (F4 @ A3)) @ ((((image @ A) @ B) @ F4) @ A6)) => ((((member @ A) @ A3) @ B6) => (((member @ A) @ A3) @ A6)))))))).
% 3.10/1.61 thf(fact_232_image__Int__subset, axiom, (! [A:$tType,B:$tType,F4:(B > A),A6:(set @ B),B6:(set @ B)]: ((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F4) @ (((inf_inf @ (set @ B)) @ A6) @ B6))) @ (((inf_inf @ (set @ A)) @ ((((image @ B) @ A) @ F4) @ A6)) @ ((((image @ B) @ A) @ F4) @ B6)))))).
% 3.10/1.61 thf(fact_233_vimage__inter__cong, axiom, (! [B:$tType,A:$tType,S4:(set @ A),F4:(A > B),G3:(A > B),Y:(set @ B)]: (((! [W:A]: (((((member @ A) @ W) @ S4) => ((F4 @ W) = (G3 @ W))))) => ((((inf_inf @ (set @ A)) @ ((((vimage @ A) @ B) @ F4) @ Y)) @ S4) = (((inf_inf @ (set @ A)) @ ((((vimage @ A) @ B) @ G3) @ Y)) @ S4)))))).
% 3.10/1.61 thf(fact_234_finite__vimage__IntI, axiom, (! [A:$tType,B:$tType,F5:(set @ A),H:(B > A),A6:(set @ B)]: ((((finite_finite2 @ A) @ F5) => (((((inj_on @ B) @ A) @ H) @ A6) => ((finite_finite2 @ B) @ (((inf_inf @ (set @ B)) @ ((((vimage @ B) @ A) @ H) @ F5)) @ A6))))))).
% 3.10/1.61 thf(fact_235_image__Int, axiom, (! [B:$tType,A:$tType,F4:(A > B),A6:(set @ A),B6:(set @ A)]: ((((((inj_on @ A) @ B) @ F4) @ (top_top @ (set @ A))) => (((((image @ A) @ B) @ F4) @ (((inf_inf @ (set @ A)) @ A6) @ B6)) = (((inf_inf @ (set @ B)) @ ((((image @ A) @ B) @ F4) @ A6)) @ ((((image @ A) @ B) @ F4) @ B6))))))).
% 3.10/1.61 thf(fact_236_inj__on__image__iff, axiom, (! [B:$tType,A:$tType,A6:(set @ A),G3:(A > B),F4:(A > A)]: (((! [X3:A]: (((((member @ A) @ X3) @ A6) => (! [Xa:A]: (((((member @ A) @ Xa) @ A6) => (((G3 @ (F4 @ X3)) = (G3 @ (F4 @ Xa))) = ((G3 @ X3) = (G3 @ Xa))))))))) => (((((inj_on @ A) @ A) @ F4) @ A6) => (((((inj_on @ A) @ B) @ G3) @ ((((image @ A) @ A) @ F4) @ A6)) = ((((inj_on @ A) @ B) @ G3) @ A6))))))).
% 3.10/1.61 thf(fact_237_inj__swap, axiom, (! [B:$tType,A:$tType,A6:(set @ ((product_prod @ A) @ B))]: (((((inj_on @ ((product_prod @ A) @ B)) @ ((product_prod @ B) @ A)) @ ((product_swap @ A) @ B)) @ A6)))).
% 3.10/1.61 thf(fact_238_Int__left__commute, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A),C4:(set @ A)]: (((((inf_inf @ (set @ A)) @ A6) @ (((inf_inf @ (set @ A)) @ B6) @ C4)) = (((inf_inf @ (set @ A)) @ B6) @ (((inf_inf @ (set @ A)) @ A6) @ C4)))))).
% 3.10/1.61 thf(fact_239_Int__left__absorb, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A)]: (((((inf_inf @ (set @ A)) @ A6) @ (((inf_inf @ (set @ A)) @ A6) @ B6)) = (((inf_inf @ (set @ A)) @ A6) @ B6))))).
% 3.10/1.61 thf(fact_240_Int__commute, axiom, (! [A:$tType]: (((inf_inf @ (set @ A)) = (^ [A7:(set @ A),B7:(set @ A)]: ((((inf_inf @ (set @ A)) @ B7) @ A7))))))).
% 3.10/1.61 thf(fact_241_Int__absorb, axiom, (! [A:$tType,A6:(set @ A)]: (((((inf_inf @ (set @ A)) @ A6) @ A6) = A6)))).
% 3.10/1.61 thf(fact_242_Int__assoc, axiom, (! [A:$tType,A6:(set @ A),B6:(set @ A),C4:(set @ A)]: (((((inf_inf @ (set @ A)) @ (((inf_inf @ (set @ A)) @ A6) @ B6)) @ C4) = (((inf_inf @ (set @ A)) @ A6) @ (((inf_inf @ (set @ A)) @ B6) @ C4)))))).
% 3.10/1.61 thf(fact_243_IntD2, axiom, (! [A:$tType,C2:A,A6:(set @ A),B6:(set @ A)]: (((((member @ A) @ C2) @ (((inf_inf @ (set @ A)) @ A6) @ B6)) => (((member @ A) @ C2) @ B6))))).
% 3.10/1.61 thf(fact_244_IntD1, axiom, (! [A:$tType,C2:A,A6:(set @ A),B6:(set @ A)]: (((((member @ A) @ C2) @ (((inf_inf @ (set @ A)) @ A6) @ B6)) => (((member @ A) @ C2) @ A6))))).
% 3.10/1.61 thf(fact_245_IntE, axiom, (! [A:$tType,C2:A,A6:(set @ A),B6:(set @ A)]: (((((member @ A) @ C2) @ (((inf_inf @ (set @ A)) @ A6) @ B6)) => (~ (((((member @ A) @ C2) @ A6) => (~ ((((member @ A) @ C2) @ B6)))))))))).
% 3.10/1.61 thf(fact_246_Int__UNIV__left, axiom, (! [A:$tType,B6:(set @ A)]: (((((inf_inf @ (set @ A)) @ (top_top @ (set @ A))) @ B6) = B6)))).
% 3.10/1.61 thf(fact_247_Int__UNIV__right, axiom, (! [A:$tType,A6:(set @ A)]: (((((inf_inf @ (set @ A)) @ A6) @ (top_top @ (set @ A))) = A6)))).
% 3.10/1.61 thf(fact_248_inj__eq, axiom, (! [B:$tType,A:$tType,F4:(A > B),X:A,Y:A]: ((((((inj_on @ A) @ B) @ F4) @ (top_top @ (set @ A))) => (((F4 @ X) = (F4 @ Y)) = (X = Y)))))).
% 3.10/1.61 thf(fact_249_injI, axiom, (! [B:$tType,A:$tType,F4:(A > B)]: (((! [X3:A,Y3:A]: ((((F4 @ X3) = (F4 @ Y3)) => (X3 = Y3)))) => ((((inj_on @ A) @ B) @ F4) @ (top_top @ (set @ A))))))).
% 3.10/1.61 thf(fact_250_injD, axiom, (! [B:$tType,A:$tType,F4:(A > B),X:A,Y:A]: ((((((inj_on @ A) @ B) @ F4) @ (top_top @ (set @ A))) => (((F4 @ X) = (F4 @ Y)) => (X = Y)))))).
% 3.10/1.61 thf(fact_251_range__ex1__eq, axiom, (! [B:$tType,A:$tType,F4:(A > B),B3:B]: ((((((inj_on @ A) @ B) @ F4) @ (top_top @ (set @ A))) => ((((member @ B) @ B3) @ ((((image @ A) @ B) @ F4) @ (top_top @ (set @ A)))) = (? [X4:A]: (((B3 = (F4 @ X4)) & (! [Y5:A]: (((B3 = (F4 @ Y5)) => (Y5 = X4)))))))))))).
% 3.10/1.61 thf(fact_252_inj__image__eq__iff, axiom, (! [B:$tType,A:$tType,F4:(A > B),A6:(set @ A),B6:(set @ A)]: ((((((inj_on @ A) @ B) @ F4) @ (top_top @ (set @ A))) => ((((((image @ A) @ B) @ F4) @ A6) = ((((image @ A) @ B) @ F4) @ B6)) = (A6 = B6)))))).
% 3.10/1.61 thf(fact_253_inj__image__mem__iff, axiom, (! [B:$tType,A:$tType,F4:(A > B),A3:A,A6:(set @ A)]: ((((((inj_on @ A) @ B) @ F4) @ (top_top @ (set @ A))) => ((((member @ B) @ (F4 @ A3)) @ ((((image @ A) @ B) @ F4) @ A6)) = (((member @ A) @ A3) @ A6)))))).
% 3.10/1.61 thf(fact_254_finite__imageD, axiom, (! [A:$tType,B:$tType,F4:(B > A),A6:(set @ B)]: ((((finite_finite2 @ A) @ ((((image @ B) @ A) @ F4) @ A6)) => (((((inj_on @ B) @ A) @ F4) @ A6) => ((finite_finite2 @ B) @ A6)))))).
% 3.10/1.61 thf(fact_255_finite__image__iff, axiom, (! [B:$tType,A:$tType,F4:(A > B),A6:(set @ A)]: ((((((inj_on @ A) @ B) @ F4) @ A6) => (((finite_finite2 @ B) @ ((((image @ A) @ B) @ F4) @ A6)) = ((finite_finite2 @ A) @ A6)))))).
% 3.10/1.61 thf(tcon_fun___Orderings_Oorder__top, axiom, (! [A8:$tType,A9:$tType]: ((((order_top @ A9) @ (type2 @ A9)) => ((order_top @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 3.10/1.61 thf(tcon_fun___Finite__Set_Ofinite, axiom, (! [A8:$tType,A9:$tType]: (((((finite_finite @ A8) @ (type2 @ A8)) & ((finite_finite @ A9) @ (type2 @ A9))) => ((finite_finite @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 3.10/1.61 thf(tcon_fun___Orderings_Otop, axiom, (! [A8:$tType,A9:$tType]: ((((top @ A9) @ (type2 @ A9)) => ((top @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 3.10/1.61 thf(tcon_fun___Orderings_Oord, axiom, (! [A8:$tType,A9:$tType]: ((((ord @ A9) @ (type2 @ A9)) => ((ord @ (A8 > A9)) @ (type2 @ (A8 > A9))))))).
% 3.10/1.61 thf(tcon_Set_Oset___Orderings_Oorder__top_1, axiom, (! [A8:$tType]: (((order_top @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 3.10/1.61 thf(tcon_Set_Oset___Finite__Set_Ofinite_2, axiom, (! [A8:$tType]: ((((finite_finite @ A8) @ (type2 @ A8)) => ((finite_finite @ (set @ A8)) @ (type2 @ (set @ A8))))))).
% 3.10/1.61 thf(tcon_Set_Oset___Orderings_Otop_3, axiom, (! [A8:$tType]: (((top @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 3.10/1.61 thf(tcon_Set_Oset___Orderings_Oord_4, axiom, (! [A8:$tType]: (((ord @ (set @ A8)) @ (type2 @ (set @ A8)))))).
% 3.10/1.61 thf(tcon_HOL_Obool___Orderings_Oorder__top_5, axiom, ((order_top @ $o) @ (type2 @ $o))).
% 3.10/1.61 thf(tcon_HOL_Obool___Finite__Set_Ofinite_6, axiom, ((finite_finite @ $o) @ (type2 @ $o))).
% 3.10/1.61 thf(tcon_HOL_Obool___Orderings_Otop_7, axiom, ((top @ $o) @ (type2 @ $o))).
% 3.10/1.61 thf(tcon_HOL_Obool___Orderings_Oord_8, axiom, ((ord @ $o) @ (type2 @ $o))).
% 3.10/1.61 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_9, axiom, (! [A8:$tType,A9:$tType]: (((((finite_finite @ A8) @ (type2 @ A8)) & ((finite_finite @ A9) @ (type2 @ A9))) => ((finite_finite @ ((sum_sum @ A8) @ A9)) @ (type2 @ ((sum_sum @ A8) @ A9))))))).
% 3.10/1.61 thf(tcon_Option_Ooption___Finite__Set_Ofinite_10, axiom, (! [A8:$tType]: ((((finite_finite @ A8) @ (type2 @ A8)) => ((finite_finite @ (option @ A8)) @ (type2 @ (option @ A8))))))).
% 3.10/1.61 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_11, axiom, (! [A8:$tType,A9:$tType]: (((((finite_finite @ A8) @ (type2 @ A8)) & ((finite_finite @ A9) @ (type2 @ A9))) => ((finite_finite @ ((product_prod @ A8) @ A9)) @ (type2 @ ((product_prod @ A8) @ A9))))))).
% 3.10/1.61 thf(conj_0, conjecture, ((root @ ((((unfold @ ((product_prod @ dtree) @ dtree)) @ parall1914194362_par_r) @ parall1914194347_par_c) @ ((((product_Pair @ dtree) @ dtree) @ tr1) @ tr2))) = ((parall1518086719_Nplus @ (root @ tr1)) @ (root @ tr2)))).
% 3.10/1.61 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.qb6mFvvipo/DTF2DTF20809.p
% 3.10/1.61 ------------------------
% 3.10/1.66 ---- Cleaned THF ---
% 3.10/1.66 thf(ty_t_Product__Type_Oprod,type,
% 3.10/1.66 product_prod: $tType > $tType > $tType ).
% 3.10/1.66
% 3.10/1.66 thf(ty_t_Sum__Type_Osum,type,
% 3.10/1.66 sum_sum: $tType > $tType > $tType ).
% 3.10/1.66
% 3.10/1.66 thf(ty_t_Option_Ooption,type,
% 3.10/1.66 option: $tType > $tType ).
% 3.10/1.66
% 3.10/1.66 thf(ty_t_DTree_Odtree,type,
% 3.10/1.66 dtree: $tType ).
% 3.10/1.66
% 3.10/1.66 thf(ty_t_Set_Oset,type,
% 3.10/1.66 set: $tType > $tType ).
% 3.10/1.66
% 3.10/1.66 thf(ty_t_DTree_OT,type,
% 3.10/1.66 t: $tType ).
% 3.10/1.66
% 3.10/1.66 thf(ty_t_DTree_ON,type,
% 3.10/1.66 n: $tType ).
% 3.10/1.66
% 3.10/1.66 thf(ty_t_itself,type,
% 3.10/1.66 itself: $tType > $tType ).
% 3.10/1.66
% 3.10/1.66 thf(sy_cl_HOL_Otype,type,
% 3.10/1.66 type:
% 3.10/1.66 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_cl_Orderings_Oord,type,
% 3.10/1.66 ord:
% 3.10/1.66 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_cl_Orderings_Otop,type,
% 3.10/1.66 top:
% 3.10/1.66 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_cl_Finite__Set_Ofinite,type,
% 3.10/1.66 finite_finite:
% 3.10/1.66 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_cl_Nat_Osemiring__char__0,type,
% 3.10/1.66 semiring_char_0:
% 3.10/1.66 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_cl_Orderings_Oorder__top,type,
% 3.10/1.66 order_top:
% 3.10/1.66 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 3.10/1.66 bNF_Ca1785829860lChain:
% 3.10/1.66 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_BNF__Def_OfstOp,type,
% 3.10/1.66 bNF_fstOp:
% 3.10/1.66 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > ( product_prod @ A @ C ) > ( product_prod @ A @ B ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_BNF__Def_Opick__middlep,type,
% 3.10/1.66 bNF_pick_middlep:
% 3.10/1.66 !>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > A ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_BNF__Def_OsndOp,type,
% 3.10/1.66 bNF_sndOp:
% 3.10/1.66 !>[C: $tType,A: $tType,B: $tType] : ( ( C > A > $o ) > ( A > B > $o ) > ( product_prod @ C @ B ) > ( product_prod @ A @ B ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
% 3.10/1.66 bNF_Greatest_image2:
% 3.10/1.66 !>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_DTree_ONode,type,
% 3.10/1.66 node: n > ( set @ ( sum_sum @ t @ dtree ) ) > dtree ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_DTree_Ocont,type,
% 3.10/1.66 cont: dtree > ( set @ ( sum_sum @ t @ dtree ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_DTree_Ocorec,type,
% 3.10/1.66 corec:
% 3.10/1.66 !>[A: $tType] : ( ( A > n ) > ( A > ( set @ ( sum_sum @ t @ ( sum_sum @ dtree @ A ) ) ) ) > A > dtree ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_DTree_Odtree_Oroot,type,
% 3.10/1.66 root: dtree > n ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_DTree_Ounfold,type,
% 3.10/1.66 unfold:
% 3.10/1.66 !>[A: $tType] : ( ( A > n ) > ( A > ( set @ ( sum_sum @ t @ A ) ) ) > A > dtree ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_Finite__Set_Ofinite,type,
% 3.10/1.66 finite_finite2:
% 3.10/1.66 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_Fun_Oinj__on,type,
% 3.10/1.66 inj_on:
% 3.10/1.66 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_Fun_Oswap,type,
% 3.10/1.66 swap:
% 3.10/1.66 !>[A: $tType,B: $tType] : ( A > A > ( A > B ) > A > B ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 3.10/1.66 inf_inf:
% 3.10/1.66 !>[A: $tType] : ( A > A > A ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 3.10/1.66 ord_less_eq:
% 3.10/1.66 !>[A: $tType] : ( A > A > $o ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_Orderings_Otop__class_Otop,type,
% 3.10/1.66 top_top:
% 3.10/1.66 !>[A: $tType] : A ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_ONplus,type,
% 3.10/1.66 parall1518086719_Nplus: n > n > n ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar,type,
% 3.10/1.66 parall1899940088le_par: ( product_prod @ dtree @ dtree ) > dtree ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__c,type,
% 3.10/1.66 parall1914194347_par_c: ( product_prod @ dtree @ dtree ) > ( set @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__r,type,
% 3.10/1.66 parall1914194362_par_r: ( product_prod @ dtree @ dtree ) > n ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_Parallel__Mirabelle__hykpkoupgu_Opar__r__rel,type,
% 3.10/1.66 parall556292031_r_rel: ( product_prod @ dtree @ dtree ) > ( product_prod @ dtree @ dtree ) > $o ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_Product__Type_OPair,type,
% 3.10/1.66 product_Pair:
% 3.10/1.66 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_Product__Type_Ocurry,type,
% 3.10/1.66 product_curry:
% 3.10/1.66 !>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 3.10/1.66 produc2004651681e_prod:
% 3.10/1.66 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_Product__Type_Omap__prod,type,
% 3.10/1.66 product_map_prod:
% 3.10/1.66 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 3.10/1.66 product_rec_prod:
% 3.10/1.66 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 3.10/1.66 product_case_prod:
% 3.10/1.66 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 3.10/1.66 product_fst:
% 3.10/1.66 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 3.10/1.66 product_snd:
% 3.10/1.66 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 3.10/1.66 product_swap:
% 3.10/1.66 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_Product__Type_Oscomp,type,
% 3.10/1.66 product_scomp:
% 3.10/1.66 !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_Pure_Otype,type,
% 3.10/1.66 type2:
% 3.10/1.66 !>[A: $tType] : ( itself @ A ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_Relation_Oinv__image,type,
% 3.10/1.66 inv_image:
% 3.10/1.66 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_Set_OCollect,type,
% 3.10/1.66 collect:
% 3.10/1.66 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_Set_OPow,type,
% 3.10/1.66 pow:
% 3.10/1.66 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_Set_Oimage,type,
% 3.10/1.66 image:
% 3.10/1.66 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_Set_Ovimage,type,
% 3.10/1.66 vimage:
% 3.10/1.66 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ B ) > ( set @ A ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_Wellfounded_Oaccp,type,
% 3.10/1.66 accp:
% 3.10/1.66 !>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_Wellfounded_Olex__prod,type,
% 3.10/1.66 lex_prod:
% 3.10/1.66 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_Wfrec_Osame__fst,type,
% 3.10/1.66 same_fst:
% 3.10/1.66 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_c_member,type,
% 3.10/1.66 member:
% 3.10/1.66 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 3.10/1.66
% 3.10/1.66 thf(sy_v_tr1,type,
% 3.10/1.66 tr1: dtree ).
% 3.10/1.66
% 3.10/1.66 thf(sy_v_tr2,type,
% 3.10/1.66 tr2: dtree ).
% 3.10/1.66
% 3.10/1.66 thf(fact_0__092_060open_062root_A_Iunfold_Apar__r_Apar__c_A_Itr1_M_Atr2_J_J_A_061_Apar__r_A_Itr1_M_Atr2_J_092_060close_062,axiom,
% 3.10/1.66 ( ( root @ ( unfold @ ( product_prod @ dtree @ dtree ) @ parall1914194362_par_r @ parall1914194347_par_c @ ( product_Pair @ dtree @ dtree @ tr1 @ tr2 ) ) )
% 3.10/1.66 = ( parall1914194362_par_r @ ( product_Pair @ dtree @ dtree @ tr1 @ tr2 ) ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_1_Nplus__comm,axiom,
% 3.10/1.66 ( parall1518086719_Nplus
% 3.10/1.66 = ( ^ [A2: n,B2: n] : ( parall1518086719_Nplus @ B2 @ A2 ) ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_2_Nplus__assoc,axiom,
% 3.10/1.66 ! [A3: n,B3: n,C2: n] :
% 3.10/1.66 ( ( parall1518086719_Nplus @ ( parall1518086719_Nplus @ A3 @ B3 ) @ C2 )
% 3.10/1.66 = ( parall1518086719_Nplus @ A3 @ ( parall1518086719_Nplus @ B3 @ C2 ) ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_3_par__r_Ocases,axiom,
% 3.10/1.66 ! [X: product_prod @ dtree @ dtree] :
% 3.10/1.66 ~ ! [Tr1: dtree,Tr2: dtree] :
% 3.10/1.66 ( X
% 3.10/1.66 != ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_4_par__r_Oelims,axiom,
% 3.10/1.66 ! [X: product_prod @ dtree @ dtree,Y: n] :
% 3.10/1.66 ( ( ( parall1914194362_par_r @ X )
% 3.10/1.66 = Y )
% 3.10/1.66 => ~ ! [Tr1: dtree,Tr2: dtree] :
% 3.10/1.66 ( ( X
% 3.10/1.66 = ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) )
% 3.10/1.66 => ( Y
% 3.10/1.66 != ( parall1518086719_Nplus @ ( root @ Tr1 ) @ ( root @ Tr2 ) ) ) ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_5_par__r_Osimps,axiom,
% 3.10/1.66 ! [Tr12: dtree,Tr22: dtree] :
% 3.10/1.66 ( ( parall1914194362_par_r @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) )
% 3.10/1.66 = ( parall1518086719_Nplus @ ( root @ Tr12 ) @ ( root @ Tr22 ) ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_6_par__r_Oinduct,axiom,
% 3.10/1.66 ! [P: ( product_prod @ dtree @ dtree ) > $o,A0: product_prod @ dtree @ dtree] :
% 3.10/1.66 ( ! [Tr1: dtree,Tr2: dtree] : ( P @ ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) )
% 3.10/1.66 => ( P @ A0 ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_7_par__def,axiom,
% 3.10/1.66 ( parall1899940088le_par
% 3.10/1.66 = ( unfold @ ( product_prod @ dtree @ dtree ) @ parall1914194362_par_r @ parall1914194347_par_c ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_8_unfold_I1_J,axiom,
% 3.10/1.66 ! [A: $tType,Rt: A > n,Ct: A > ( set @ ( sum_sum @ t @ A ) ),B3: A] :
% 3.10/1.66 ( ( root @ ( unfold @ A @ Rt @ Ct @ B3 ) )
% 3.10/1.66 = ( Rt @ B3 ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_9_prod_Oinject,axiom,
% 3.10/1.66 ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
% 3.10/1.66 ( ( ( product_Pair @ A @ B @ X1 @ X2 )
% 3.10/1.66 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 3.10/1.66 = ( ( X1 = Y1 )
% 3.10/1.66 & ( X2 = Y2 ) ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_10_old_Oprod_Oinject,axiom,
% 3.10/1.66 ! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
% 3.10/1.66 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 3.10/1.66 = ( product_Pair @ A @ B @ A4 @ B4 ) )
% 3.10/1.66 = ( ( A3 = A4 )
% 3.10/1.66 & ( B3 = B4 ) ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_11_par__r_Opelims,axiom,
% 3.10/1.66 ! [X: product_prod @ dtree @ dtree,Y: n] :
% 3.10/1.66 ( ( ( parall1914194362_par_r @ X )
% 3.10/1.66 = Y )
% 3.10/1.66 => ( ( accp @ ( product_prod @ dtree @ dtree ) @ parall556292031_r_rel @ X )
% 3.10/1.66 => ~ ! [Tr1: dtree,Tr2: dtree] :
% 3.10/1.66 ( ( X
% 3.10/1.66 = ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) )
% 3.10/1.66 => ( ( Y
% 3.10/1.66 = ( parall1518086719_Nplus @ ( root @ Tr1 ) @ ( root @ Tr2 ) ) )
% 3.10/1.66 => ~ ( accp @ ( product_prod @ dtree @ dtree ) @ parall556292031_r_rel @ ( product_Pair @ dtree @ dtree @ Tr1 @ Tr2 ) ) ) ) ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_12_surj__pair,axiom,
% 3.10/1.66 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 3.10/1.66 ? [X3: A,Y3: B] :
% 3.10/1.66 ( P2
% 3.10/1.66 = ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_13_prod__cases,axiom,
% 3.10/1.66 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 3.10/1.66 ( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 3.10/1.66 => ( P @ P2 ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_14_Pair__inject,axiom,
% 3.10/1.66 ! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
% 3.10/1.66 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 3.10/1.66 = ( product_Pair @ A @ B @ A4 @ B4 ) )
% 3.10/1.66 => ~ ( ( A3 = A4 )
% 3.10/1.66 => ( B3 != B4 ) ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_15_prod__cases3,axiom,
% 3.10/1.66 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 3.10/1.66 ~ ! [A5: A,B5: B,C3: C] :
% 3.10/1.66 ( Y
% 3.10/1.66 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_16_prod__cases4,axiom,
% 3.10/1.66 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 3.10/1.66 ~ ! [A5: A,B5: B,C3: C,D2: D] :
% 3.10/1.66 ( Y
% 3.10/1.66 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_17_prod__cases5,axiom,
% 3.10/1.66 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 3.10/1.66 ~ ! [A5: A,B5: B,C3: C,D2: D,E2: E] :
% 3.10/1.66 ( Y
% 3.10/1.66 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_18_prod__cases6,axiom,
% 3.10/1.66 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 3.10/1.66 ~ ! [A5: A,B5: B,C3: C,D2: D,E2: E,F2: F] :
% 3.10/1.66 ( Y
% 3.10/1.66 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_19_old_Oprod_Oinducts,axiom,
% 3.10/1.66 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 3.10/1.66 ( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 3.10/1.66 => ( P @ Prod ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_20_old_Oprod_Oexhaust,axiom,
% 3.10/1.66 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 3.10/1.66 ~ ! [A5: A,B5: B] :
% 3.10/1.66 ( Y
% 3.10/1.66 != ( product_Pair @ A @ B @ A5 @ B5 ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_21_prod__induct7,axiom,
% 3.10/1.66 ! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 3.10/1.66 ( ! [A5: A,B5: B,C3: C,D2: D,E2: E,F2: F,G2: G] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
% 3.10/1.66 => ( P @ X ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_22_prod__induct6,axiom,
% 3.10/1.66 ! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 3.10/1.66 ( ! [A5: A,B5: B,C3: C,D2: D,E2: E,F2: F] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
% 3.10/1.66 => ( P @ X ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_23_prod__induct5,axiom,
% 3.10/1.66 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 3.10/1.66 ( ! [A5: A,B5: B,C3: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 3.10/1.66 => ( P @ X ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_24_prod__induct4,axiom,
% 3.10/1.66 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 3.10/1.66 ( ! [A5: A,B5: B,C3: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
% 3.10/1.66 => ( P @ X ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_25_prod__induct3,axiom,
% 3.10/1.66 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 3.10/1.66 ( ! [A5: A,B5: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) )
% 3.10/1.66 => ( P @ X ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_26_prod__cases7,axiom,
% 3.10/1.66 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 3.10/1.66 ~ ! [A5: A,B5: B,C3: C,D2: D,E2: E,F2: F,G2: G] :
% 3.10/1.66 ( Y
% 3.10/1.66 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_27_old_Oprod_Orec,axiom,
% 3.10/1.66 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B3: B] :
% 3.10/1.66 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 3.10/1.66 = ( F1 @ A3 @ B3 ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_28_internal__case__prod__conv,axiom,
% 3.10/1.66 ! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A3: B,B3: C] :
% 3.10/1.66 ( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A3 @ B3 ) )
% 3.10/1.66 = ( C2 @ A3 @ B3 ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_29_root__Node,axiom,
% 3.10/1.66 ! [N: n,As: set @ ( sum_sum @ t @ dtree )] :
% 3.10/1.66 ( ( root @ ( node @ N @ As ) )
% 3.10/1.66 = N ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_30_accp_Ocases,axiom,
% 3.10/1.66 ! [A: $tType,R: A > A > $o,A3: A] :
% 3.10/1.66 ( ( accp @ A @ R @ A3 )
% 3.10/1.66 => ! [Y4: A] :
% 3.10/1.66 ( ( R @ Y4 @ A3 )
% 3.10/1.66 => ( accp @ A @ R @ Y4 ) ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_31_accp_Osimps,axiom,
% 3.10/1.66 ! [A: $tType] :
% 3.10/1.66 ( ( accp @ A )
% 3.10/1.66 = ( ^ [R2: A > A > $o,A2: A] :
% 3.10/1.66 ? [X4: A] :
% 3.10/1.66 ( ( A2 = X4 )
% 3.10/1.66 & ! [Y5: A] :
% 3.10/1.66 ( ( R2 @ Y5 @ X4 )
% 3.10/1.66 => ( accp @ A @ R2 @ Y5 ) ) ) ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_32_accp_Ointros,axiom,
% 3.10/1.66 ! [A: $tType,R: A > A > $o,X: A] :
% 3.10/1.66 ( ! [Y3: A] :
% 3.10/1.66 ( ( R @ Y3 @ X )
% 3.10/1.66 => ( accp @ A @ R @ Y3 ) )
% 3.10/1.66 => ( accp @ A @ R @ X ) ) ).
% 3.10/1.66
% 3.10/1.66 thf(fact_33_accp__induct,axiom,
% 3.10/1.66 ! [A: $tType,R: A > A > $o,A3: A,P: A > $o] :
% 3.10/1.66 ( ( accp @ A @ R @ A3 )
% 3.10/1.66 => ( ! [X3: A] :
% 3.10/1.66 ( ( accp @ A @ R @ X3 )
% 3.10/1.66 => ( ! [Y4: A] :
% 3.10/1.66 ( ( R @ Y4 @ X3 )
% 3.10/1.66 => ( P @ Y4 ) )
% 3.10/1.67 => ( P @ X3 ) ) )
% 3.10/1.67 => ( P @ A3 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_34_accp_Oinducts,axiom,
% 3.10/1.67 ! [A: $tType,R: A > A > $o,X: A,P: A > $o] :
% 3.10/1.67 ( ( accp @ A @ R @ X )
% 3.10/1.67 => ( ! [X3: A] :
% 3.10/1.67 ( ! [Y4: A] :
% 3.10/1.67 ( ( R @ Y4 @ X3 )
% 3.10/1.67 => ( accp @ A @ R @ Y4 ) )
% 3.10/1.67 => ( ! [Y4: A] :
% 3.10/1.67 ( ( R @ Y4 @ X3 )
% 3.10/1.67 => ( P @ Y4 ) )
% 3.10/1.67 => ( P @ X3 ) ) )
% 3.10/1.67 => ( P @ X ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_35_accp__downward,axiom,
% 3.10/1.67 ! [A: $tType,R: A > A > $o,B3: A,A3: A] :
% 3.10/1.67 ( ( accp @ A @ R @ B3 )
% 3.10/1.67 => ( ( R @ A3 @ B3 )
% 3.10/1.67 => ( accp @ A @ R @ A3 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_36_not__accp__down,axiom,
% 3.10/1.67 ! [A: $tType,R3: A > A > $o,X: A] :
% 3.10/1.67 ( ~ ( accp @ A @ R3 @ X )
% 3.10/1.67 => ~ ! [Z: A] :
% 3.10/1.67 ( ( R3 @ Z @ X )
% 3.10/1.67 => ( accp @ A @ R3 @ Z ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_37_accp__induct__rule,axiom,
% 3.10/1.67 ! [A: $tType,R: A > A > $o,A3: A,P: A > $o] :
% 3.10/1.67 ( ( accp @ A @ R @ A3 )
% 3.10/1.67 => ( ! [X3: A] :
% 3.10/1.67 ( ( accp @ A @ R @ X3 )
% 3.10/1.67 => ( ! [Y4: A] :
% 3.10/1.67 ( ( R @ Y4 @ X3 )
% 3.10/1.67 => ( P @ Y4 ) )
% 3.10/1.67 => ( P @ X3 ) ) )
% 3.10/1.67 => ( P @ A3 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_38_in__lex__prod,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] :
% 3.10/1.67 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Pair @ A @ B @ A4 @ B4 ) ) @ ( lex_prod @ A @ B @ R @ S ) )
% 3.10/1.67 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A4 ) @ R )
% 3.10/1.67 | ( ( A3 = A4 )
% 3.10/1.67 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B3 @ B4 ) @ S ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_39_Node__root__cont,axiom,
% 3.10/1.67 ! [Tr: dtree] :
% 3.10/1.67 ( ( node @ ( root @ Tr ) @ ( cont @ Tr ) )
% 3.10/1.67 = Tr ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_40_corec_I1_J,axiom,
% 3.10/1.67 ! [A: $tType,Rt: A > n,Ct: A > ( set @ ( sum_sum @ t @ ( sum_sum @ dtree @ A ) ) ),B3: A] :
% 3.10/1.67 ( ( root @ ( corec @ A @ Rt @ Ct @ B3 ) )
% 3.10/1.67 = ( Rt @ B3 ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_41_ssubst__Pair__rhs,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,R: A,S: B,R3: set @ ( product_prod @ A @ B ),S2: B] :
% 3.10/1.67 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S ) @ R3 )
% 3.10/1.67 => ( ( S2 = S )
% 3.10/1.67 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S2 ) @ R3 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_42_curry__conv,axiom,
% 3.10/1.67 ! [A: $tType,C: $tType,B: $tType] :
% 3.10/1.67 ( ( product_curry @ B @ C @ A )
% 3.10/1.67 = ( ^ [F3: ( product_prod @ B @ C ) > A,A2: B,B2: C] : ( F3 @ ( product_Pair @ B @ C @ A2 @ B2 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_43_finite__par__c,axiom,
% 3.10/1.67 ! [Tr12: dtree,Tr22: dtree] : ( finite_finite2 @ ( sum_sum @ t @ ( product_prod @ dtree @ dtree ) ) @ ( parall1914194347_par_c @ ( product_Pair @ dtree @ dtree @ Tr12 @ Tr22 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_44_curryI,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,F4: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
% 3.10/1.67 ( ( F4 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 3.10/1.67 => ( product_curry @ A @ B @ $o @ F4 @ A3 @ B3 ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_45_mem__Collect__eq,axiom,
% 3.10/1.67 ! [A: $tType,A3: A,P: A > $o] :
% 3.10/1.67 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 3.10/1.67 = ( P @ A3 ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_46_Collect__mem__eq,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A] :
% 3.10/1.67 ( ( collect @ A
% 3.10/1.67 @ ^ [X4: A] : ( member @ A @ X4 @ A6 ) )
% 3.10/1.67 = A6 ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_47_Collect__cong,axiom,
% 3.10/1.67 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.10/1.67 ( ! [X3: A] :
% 3.10/1.67 ( ( P @ X3 )
% 3.10/1.67 = ( Q @ X3 ) )
% 3.10/1.67 => ( ( collect @ A @ P )
% 3.10/1.67 = ( collect @ A @ Q ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_48_ext,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,F4: A > B,G3: A > B] :
% 3.10/1.67 ( ! [X3: A] :
% 3.10/1.67 ( ( F4 @ X3 )
% 3.10/1.67 = ( G3 @ X3 ) )
% 3.10/1.67 => ( F4 = G3 ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_49_swap__simp,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,X: B,Y: A] :
% 3.10/1.67 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 3.10/1.67 = ( product_Pair @ A @ B @ Y @ X ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_50_internal__case__prod__def,axiom,
% 3.10/1.67 ! [C: $tType,B: $tType,A: $tType] :
% 3.10/1.67 ( ( produc2004651681e_prod @ A @ B @ C )
% 3.10/1.67 = ( product_case_prod @ A @ B @ C ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_51_swap__swap,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 3.10/1.67 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
% 3.10/1.67 = P2 ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_52_curry__case__prod,axiom,
% 3.10/1.67 ! [C: $tType,B: $tType,A: $tType,F4: A > B > C] :
% 3.10/1.67 ( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F4 ) )
% 3.10/1.67 = F4 ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_53_case__prod__curry,axiom,
% 3.10/1.67 ! [C: $tType,B: $tType,A: $tType,F4: ( product_prod @ A @ B ) > C] :
% 3.10/1.67 ( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F4 ) )
% 3.10/1.67 = F4 ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_54_mem__case__prodE,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),P2: product_prod @ B @ C] :
% 3.10/1.67 ( ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P2 ) )
% 3.10/1.67 => ~ ! [X3: B,Y3: C] :
% 3.10/1.67 ( ( P2
% 3.10/1.67 = ( product_Pair @ B @ C @ X3 @ Y3 ) )
% 3.10/1.67 => ~ ( member @ A @ Z2 @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_55_old_Oprod_Ocase,axiom,
% 3.10/1.67 ! [A: $tType,C: $tType,B: $tType,F4: A > B > C,X1: A,X2: B] :
% 3.10/1.67 ( ( product_case_prod @ A @ B @ C @ F4 @ ( product_Pair @ A @ B @ X1 @ X2 ) )
% 3.10/1.67 = ( F4 @ X1 @ X2 ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_56_dtree__cong,axiom,
% 3.10/1.67 ! [Tr: dtree,Tr3: dtree] :
% 3.10/1.67 ( ( ( root @ Tr )
% 3.10/1.67 = ( root @ Tr3 ) )
% 3.10/1.67 => ( ( ( cont @ Tr )
% 3.10/1.67 = ( cont @ Tr3 ) )
% 3.10/1.67 => ( Tr = Tr3 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_57_curryE,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,F4: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
% 3.10/1.67 ( ( product_curry @ A @ B @ $o @ F4 @ A3 @ B3 )
% 3.10/1.67 => ( F4 @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_58_curryD,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,F4: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
% 3.10/1.67 ( ( product_curry @ A @ B @ $o @ F4 @ A3 @ B3 )
% 3.10/1.67 => ( F4 @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_59_finite__code,axiom,
% 3.10/1.67 ! [A: $tType] :
% 3.10/1.67 ( ( finite_finite @ A @ ( type2 @ A ) )
% 3.10/1.67 => ( ( finite_finite2 @ A )
% 3.10/1.67 = ( ^ [A7: set @ A] : $true ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_60_split__cong,axiom,
% 3.10/1.67 ! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F4: A > B > C,G3: A > B > C,P2: product_prod @ A @ B] :
% 3.10/1.67 ( ! [X3: A,Y3: B] :
% 3.10/1.67 ( ( ( product_Pair @ A @ B @ X3 @ Y3 )
% 3.10/1.67 = Q2 )
% 3.10/1.67 => ( ( F4 @ X3 @ Y3 )
% 3.10/1.67 = ( G3 @ X3 @ Y3 ) ) )
% 3.10/1.67 => ( ( P2 = Q2 )
% 3.10/1.67 => ( ( product_case_prod @ A @ B @ C @ F4 @ P2 )
% 3.10/1.67 = ( product_case_prod @ A @ B @ C @ G3 @ Q2 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_61_same__fstI,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,P: A > $o,X: A,Y6: B,Y: B,R3: A > ( set @ ( product_prod @ B @ B ) )] :
% 3.10/1.67 ( ( P @ X )
% 3.10/1.67 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y6 @ Y ) @ ( R3 @ X ) )
% 3.10/1.67 => ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y6 ) @ ( product_Pair @ A @ B @ X @ Y ) ) @ ( same_fst @ A @ B @ P @ R3 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_62_cont__Node,axiom,
% 3.10/1.67 ! [As: set @ ( sum_sum @ t @ dtree ),N: n] :
% 3.10/1.67 ( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As )
% 3.10/1.67 => ( ( cont @ ( node @ N @ As ) )
% 3.10/1.67 = As ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_63_finite__set__choice,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,A6: set @ A,P: A > B > $o] :
% 3.10/1.67 ( ( finite_finite2 @ A @ A6 )
% 3.10/1.67 => ( ! [X3: A] :
% 3.10/1.67 ( ( member @ A @ X3 @ A6 )
% 3.10/1.67 => ? [X12: B] : ( P @ X3 @ X12 ) )
% 3.10/1.67 => ? [F2: A > B] :
% 3.10/1.67 ! [X5: A] :
% 3.10/1.67 ( ( member @ A @ X5 @ A6 )
% 3.10/1.67 => ( P @ X5 @ ( F2 @ X5 ) ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_64_finite,axiom,
% 3.10/1.67 ! [A: $tType] :
% 3.10/1.67 ( ( finite_finite @ A @ ( type2 @ A ) )
% 3.10/1.67 => ! [A6: set @ A] : ( finite_finite2 @ A @ A6 ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_65_scomp__apply,axiom,
% 3.10/1.67 ! [A: $tType,D: $tType,C: $tType,B: $tType] :
% 3.10/1.67 ( ( product_scomp @ B @ C @ D @ A )
% 3.10/1.67 = ( ^ [F3: B > ( product_prod @ C @ D ),G4: C > D > A,X4: B] : ( product_case_prod @ C @ D @ A @ G4 @ ( F3 @ X4 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_66_pair__in__swap__image,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,Y: A,X: B,A6: set @ ( product_prod @ B @ A )] :
% 3.10/1.67 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A6 ) )
% 3.10/1.67 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A6 ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_67_finite__imageI,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,F5: set @ A,H: A > B] :
% 3.10/1.67 ( ( finite_finite2 @ A @ F5 )
% 3.10/1.67 => ( finite_finite2 @ B @ ( image @ A @ B @ H @ F5 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_68_dtree__simps,axiom,
% 3.10/1.67 ! [As: set @ ( sum_sum @ t @ dtree ),As2: set @ ( sum_sum @ t @ dtree ),N: n,N2: n] :
% 3.10/1.67 ( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As )
% 3.10/1.67 => ( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As2 )
% 3.10/1.67 => ( ( ( node @ N @ As )
% 3.10/1.67 = ( node @ N2 @ As2 ) )
% 3.10/1.67 = ( ( N = N2 )
% 3.10/1.67 & ( As = As2 ) ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_69_finite__cont,axiom,
% 3.10/1.67 ! [Tr: dtree] : ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ ( cont @ Tr ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_70_dtree__cases,axiom,
% 3.10/1.67 ! [Tr: dtree] :
% 3.10/1.67 ~ ! [N3: n,As3: set @ ( sum_sum @ t @ dtree )] :
% 3.10/1.67 ( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As3 )
% 3.10/1.67 => ( Tr
% 3.10/1.67 != ( node @ N3 @ As3 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_71_scomp__Pair,axiom,
% 3.10/1.67 ! [C: $tType,B: $tType,A: $tType,X: A > ( product_prod @ B @ C )] :
% 3.10/1.67 ( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X @ ( product_Pair @ B @ C ) )
% 3.10/1.67 = X ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_72_Pair__scomp,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,C: $tType,X: C,F4: C > A > B] :
% 3.10/1.67 ( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X ) @ F4 )
% 3.10/1.67 = ( F4 @ X ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_73_scomp__def,axiom,
% 3.10/1.67 ! [D: $tType,C: $tType,B: $tType,A: $tType] :
% 3.10/1.67 ( ( product_scomp @ A @ B @ C @ D )
% 3.10/1.67 = ( ^ [F3: A > ( product_prod @ B @ C ),G4: B > C > D,X4: A] : ( product_case_prod @ B @ C @ D @ G4 @ ( F3 @ X4 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_74_image__eqI,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,B3: A,F4: B > A,X: B,A6: set @ B] :
% 3.10/1.67 ( ( B3
% 3.10/1.67 = ( F4 @ X ) )
% 3.10/1.67 => ( ( member @ B @ X @ A6 )
% 3.10/1.67 => ( member @ A @ B3 @ ( image @ B @ A @ F4 @ A6 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_75_surj__swap,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType] :
% 3.10/1.67 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 3.10/1.67 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_76_Sup_OSUP__cong,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,A6: set @ B,B6: set @ B,C4: B > A,D3: B > A,Sup: ( set @ A ) > A] :
% 3.10/1.67 ( ( A6 = B6 )
% 3.10/1.67 => ( ! [X3: B] :
% 3.10/1.67 ( ( member @ B @ X3 @ B6 )
% 3.10/1.67 => ( ( C4 @ X3 )
% 3.10/1.67 = ( D3 @ X3 ) ) )
% 3.10/1.67 => ( ( Sup @ ( image @ B @ A @ C4 @ A6 ) )
% 3.10/1.67 = ( Sup @ ( image @ B @ A @ D3 @ B6 ) ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_77_Inf_OINF__cong,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,A6: set @ B,B6: set @ B,C4: B > A,D3: B > A,Inf: ( set @ A ) > A] :
% 3.10/1.67 ( ( A6 = B6 )
% 3.10/1.67 => ( ! [X3: B] :
% 3.10/1.67 ( ( member @ B @ X3 @ B6 )
% 3.10/1.67 => ( ( C4 @ X3 )
% 3.10/1.67 = ( D3 @ X3 ) ) )
% 3.10/1.67 => ( ( Inf @ ( image @ B @ A @ C4 @ A6 ) )
% 3.10/1.67 = ( Inf @ ( image @ B @ A @ D3 @ B6 ) ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_78_rev__image__eqI,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,X: A,A6: set @ A,B3: B,F4: A > B] :
% 3.10/1.67 ( ( member @ A @ X @ A6 )
% 3.10/1.67 => ( ( B3
% 3.10/1.67 = ( F4 @ X ) )
% 3.10/1.67 => ( member @ B @ B3 @ ( image @ A @ B @ F4 @ A6 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_79_ball__imageD,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B,P: A > $o] :
% 3.10/1.67 ( ! [X3: A] :
% 3.10/1.67 ( ( member @ A @ X3 @ ( image @ B @ A @ F4 @ A6 ) )
% 3.10/1.67 => ( P @ X3 ) )
% 3.10/1.67 => ! [X5: B] :
% 3.10/1.67 ( ( member @ B @ X5 @ A6 )
% 3.10/1.67 => ( P @ ( F4 @ X5 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_80_image__cong,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,M: set @ A,N4: set @ A,F4: A > B,G3: A > B] :
% 3.10/1.67 ( ( M = N4 )
% 3.10/1.67 => ( ! [X3: A] :
% 3.10/1.67 ( ( member @ A @ X3 @ N4 )
% 3.10/1.67 => ( ( F4 @ X3 )
% 3.10/1.67 = ( G3 @ X3 ) ) )
% 3.10/1.67 => ( ( image @ A @ B @ F4 @ M )
% 3.10/1.67 = ( image @ A @ B @ G3 @ N4 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_81_UNIV__I,axiom,
% 3.10/1.67 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_82_finite__Plus__UNIV__iff,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType] :
% 3.10/1.67 ( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) )
% 3.10/1.67 = ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
% 3.10/1.67 & ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_83_Finite__Set_Ofinite__set,axiom,
% 3.10/1.67 ! [A: $tType] :
% 3.10/1.67 ( ( finite_finite2 @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
% 3.10/1.67 = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_84_finite__prod,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType] :
% 3.10/1.67 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 3.10/1.67 = ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
% 3.10/1.67 & ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_85_finite__Prod__UNIV,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType] :
% 3.10/1.67 ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
% 3.10/1.67 => ( ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) )
% 3.10/1.67 => ( finite_finite2 @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_86_finite__fun__UNIVD2,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType] :
% 3.10/1.67 ( ( finite_finite2 @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) )
% 3.10/1.67 => ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_87_range__eqI,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,B3: A,F4: B > A,X: B] :
% 3.10/1.67 ( ( B3
% 3.10/1.67 = ( F4 @ X ) )
% 3.10/1.67 => ( member @ A @ B3 @ ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_88_rangeI,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,F4: B > A,X: B] : ( member @ A @ ( F4 @ X ) @ ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_89_UNIV__eq__I,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A] :
% 3.10/1.67 ( ! [X3: A] : ( member @ A @ X3 @ A6 )
% 3.10/1.67 => ( ( top_top @ ( set @ A ) )
% 3.10/1.67 = A6 ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_90_UNIV__witness,axiom,
% 3.10/1.67 ! [A: $tType] :
% 3.10/1.67 ? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_91_finite__UNIV,axiom,
% 3.10/1.67 ! [A: $tType] :
% 3.10/1.67 ( ( finite_finite @ A @ ( type2 @ A ) )
% 3.10/1.67 => ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_92_infinite__UNIV__char__0,axiom,
% 3.10/1.67 ! [A: $tType] :
% 3.10/1.67 ( ( semiring_char_0 @ A @ ( type2 @ A ) )
% 3.10/1.67 => ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_93_ex__new__if__finite,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A] :
% 3.10/1.67 ( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
% 3.10/1.67 => ( ( finite_finite2 @ A @ A6 )
% 3.10/1.67 => ? [A5: A] :
% 3.10/1.67 ~ ( member @ A @ A5 @ A6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_94_imageI,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,X: A,A6: set @ A,F4: A > B] :
% 3.10/1.67 ( ( member @ A @ X @ A6 )
% 3.10/1.67 => ( member @ B @ ( F4 @ X ) @ ( image @ A @ B @ F4 @ A6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_95_image__iff,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,Z2: A,F4: B > A,A6: set @ B] :
% 3.10/1.67 ( ( member @ A @ Z2 @ ( image @ B @ A @ F4 @ A6 ) )
% 3.10/1.67 = ( ? [X4: B] :
% 3.10/1.67 ( ( member @ B @ X4 @ A6 )
% 3.10/1.67 & ( Z2
% 3.10/1.67 = ( F4 @ X4 ) ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_96_bex__imageD,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B,P: A > $o] :
% 3.10/1.67 ( ? [X5: A] :
% 3.10/1.67 ( ( member @ A @ X5 @ ( image @ B @ A @ F4 @ A6 ) )
% 3.10/1.67 & ( P @ X5 ) )
% 3.10/1.67 => ? [X3: B] :
% 3.10/1.67 ( ( member @ B @ X3 @ A6 )
% 3.10/1.67 & ( P @ ( F4 @ X3 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_97_finite__option__UNIV,axiom,
% 3.10/1.67 ! [A: $tType] :
% 3.10/1.67 ( ( finite_finite2 @ ( option @ A ) @ ( top_top @ ( set @ ( option @ A ) ) ) )
% 3.10/1.67 = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_98_iso__tuple__UNIV__I,axiom,
% 3.10/1.67 ! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_99_top__apply,axiom,
% 3.10/1.67 ! [C: $tType,D: $tType] :
% 3.10/1.67 ( ( top @ C @ ( type2 @ C ) )
% 3.10/1.67 => ( ( top_top @ ( D > C ) )
% 3.10/1.67 = ( ^ [X4: D] : ( top_top @ C ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_100_surj__def,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,F4: B > A] :
% 3.10/1.67 ( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 3.10/1.67 = ( top_top @ ( set @ A ) ) )
% 3.10/1.67 = ( ! [Y5: A] :
% 3.10/1.67 ? [X4: B] :
% 3.10/1.67 ( Y5
% 3.10/1.67 = ( F4 @ X4 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_101_surjI,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,G3: B > A,F4: A > B] :
% 3.10/1.67 ( ! [X3: A] :
% 3.10/1.67 ( ( G3 @ ( F4 @ X3 ) )
% 3.10/1.67 = X3 )
% 3.10/1.67 => ( ( image @ B @ A @ G3 @ ( top_top @ ( set @ B ) ) )
% 3.10/1.67 = ( top_top @ ( set @ A ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_102_surjE,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,F4: B > A,Y: A] :
% 3.10/1.67 ( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 3.10/1.67 = ( top_top @ ( set @ A ) ) )
% 3.10/1.67 => ~ ! [X3: B] :
% 3.10/1.67 ( Y
% 3.10/1.67 != ( F4 @ X3 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_103_top__set__def,axiom,
% 3.10/1.67 ! [A: $tType] :
% 3.10/1.67 ( ( top_top @ ( set @ A ) )
% 3.10/1.67 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_104_surjD,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,F4: B > A,Y: A] :
% 3.10/1.67 ( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 3.10/1.67 = ( top_top @ ( set @ A ) ) )
% 3.10/1.67 => ? [X3: B] :
% 3.10/1.67 ( Y
% 3.10/1.67 = ( F4 @ X3 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_105_map__prod__surj,axiom,
% 3.10/1.67 ! [A: $tType,C: $tType,D: $tType,B: $tType,F4: A > B,G3: C > D] :
% 3.10/1.67 ( ( ( image @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 3.10/1.67 = ( top_top @ ( set @ B ) ) )
% 3.10/1.67 => ( ( ( image @ C @ D @ G3 @ ( top_top @ ( set @ C ) ) )
% 3.10/1.67 = ( top_top @ ( set @ D ) ) )
% 3.10/1.67 => ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F4 @ G3 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
% 3.10/1.67 = ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_106_Pow__UNIV,axiom,
% 3.10/1.67 ! [A: $tType] :
% 3.10/1.67 ( ( pow @ A @ ( top_top @ ( set @ A ) ) )
% 3.10/1.67 = ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_107_surj__swap__iff,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,A3: B,B3: B,F4: B > A] :
% 3.10/1.67 ( ( ( image @ B @ A @ ( swap @ B @ A @ A3 @ B3 @ F4 ) @ ( top_top @ ( set @ B ) ) )
% 3.10/1.67 = ( top_top @ ( set @ A ) ) )
% 3.10/1.67 = ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 3.10/1.67 = ( top_top @ ( set @ A ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_108_top__empty__eq,axiom,
% 3.10/1.67 ! [A: $tType] :
% 3.10/1.67 ( ( top_top @ ( A > $o ) )
% 3.10/1.67 = ( ^ [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_109_finite__Pow__iff,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A] :
% 3.10/1.67 ( ( finite_finite2 @ ( set @ A ) @ ( pow @ A @ A6 ) )
% 3.10/1.67 = ( finite_finite2 @ A @ A6 ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_110_top1I,axiom,
% 3.10/1.67 ! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_111_map__prod__simp,axiom,
% 3.10/1.67 ! [C: $tType,A: $tType,B: $tType,D: $tType,F4: C > A,G3: D > B,A3: C,B3: D] :
% 3.10/1.67 ( ( product_map_prod @ C @ A @ D @ B @ F4 @ G3 @ ( product_Pair @ C @ D @ A3 @ B3 ) )
% 3.10/1.67 = ( product_Pair @ A @ B @ ( F4 @ A3 ) @ ( G3 @ B3 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_112_swap__image__eq,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,A3: A,A6: set @ A,B3: A,F4: A > B] :
% 3.10/1.67 ( ( member @ A @ A3 @ A6 )
% 3.10/1.67 => ( ( member @ A @ B3 @ A6 )
% 3.10/1.67 => ( ( image @ A @ B @ ( swap @ A @ B @ A3 @ B3 @ F4 ) @ A6 )
% 3.10/1.67 = ( image @ A @ B @ F4 @ A6 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_113_map__prod__imageI,axiom,
% 3.10/1.67 ! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B3: B,R3: set @ ( product_prod @ A @ B ),F4: A > C,G3: B > D] :
% 3.10/1.67 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R3 )
% 3.10/1.67 => ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F4 @ A3 ) @ ( G3 @ B3 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F4 @ G3 ) @ R3 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_114_Cantors__paradox,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A] :
% 3.10/1.67 ~ ? [X5: A > ( set @ A )] :
% 3.10/1.67 ( ( image @ A @ ( set @ A ) @ X5 @ A6 )
% 3.10/1.67 = ( pow @ A @ A6 ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_115_Pow__top,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A] : ( member @ ( set @ A ) @ A6 @ ( pow @ A @ A6 ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_116_image__Pow__surj,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,F4: B > A,A6: set @ B,B6: set @ A] :
% 3.10/1.67 ( ( ( image @ B @ A @ F4 @ A6 )
% 3.10/1.67 = B6 )
% 3.10/1.67 => ( ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F4 ) @ ( pow @ B @ A6 ) )
% 3.10/1.67 = ( pow @ A @ B6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_117_prod__fun__imageE,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,D: $tType,C: $tType,C2: product_prod @ A @ B,F4: C > A,G3: D > B,R3: set @ ( product_prod @ C @ D )] :
% 3.10/1.67 ( ( member @ ( product_prod @ A @ B ) @ C2 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F4 @ G3 ) @ R3 ) )
% 3.10/1.67 => ~ ! [X3: C,Y3: D] :
% 3.10/1.67 ( ( C2
% 3.10/1.67 = ( product_Pair @ A @ B @ ( F4 @ X3 ) @ ( G3 @ Y3 ) ) )
% 3.10/1.67 => ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X3 @ Y3 ) @ R3 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_118_surj__imp__surj__swap,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,F4: B > A,A3: B,B3: B] :
% 3.10/1.67 ( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 3.10/1.67 = ( top_top @ ( set @ A ) ) )
% 3.10/1.67 => ( ( image @ B @ A @ ( swap @ B @ A @ A3 @ B3 @ F4 ) @ ( top_top @ ( set @ B ) ) )
% 3.10/1.67 = ( top_top @ ( set @ A ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_119_top__conj_I2_J,axiom,
% 3.10/1.67 ! [A: $tType,P: $o,X: A] :
% 3.10/1.67 ( ( P
% 3.10/1.67 & ( top_top @ ( A > $o ) @ X ) )
% 3.10/1.67 = P ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_120_top__conj_I1_J,axiom,
% 3.10/1.67 ! [A: $tType,X: A,P: $o] :
% 3.10/1.67 ( ( ( top_top @ ( A > $o ) @ X )
% 3.10/1.67 & P )
% 3.10/1.67 = P ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_121_range__fst,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType] :
% 3.10/1.67 ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 3.10/1.67 = ( top_top @ ( set @ A ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_122_range__snd,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType] :
% 3.10/1.67 ( ( image @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 3.10/1.67 = ( top_top @ ( set @ A ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_123_in__inv__image,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,X: A,Y: A,R: set @ ( product_prod @ B @ B ),F4: A > B] :
% 3.10/1.67 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( inv_image @ B @ A @ R @ F4 ) )
% 3.10/1.67 = ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F4 @ X ) @ ( F4 @ Y ) ) @ R ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_124_fst__map__prod,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,D: $tType,C: $tType,F4: C > A,G3: D > B,X: product_prod @ C @ D] :
% 3.10/1.67 ( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F4 @ G3 @ X ) )
% 3.10/1.67 = ( F4 @ ( product_fst @ C @ D @ X ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_125_snd__map__prod,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,D: $tType,C: $tType,F4: C > B,G3: D > A,X: product_prod @ C @ D] :
% 3.10/1.67 ( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F4 @ G3 @ X ) )
% 3.10/1.67 = ( G3 @ ( product_snd @ C @ D @ X ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_126_prod_Ocollapse,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 3.10/1.67 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 3.10/1.67 = Prod ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_127_snd__swap,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,X: product_prod @ A @ B] :
% 3.10/1.67 ( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
% 3.10/1.67 = ( product_fst @ A @ B @ X ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_128_fst__swap,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,X: product_prod @ B @ A] :
% 3.10/1.67 ( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
% 3.10/1.67 = ( product_snd @ B @ A @ X ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_129_prod_Osplit__sel,axiom,
% 3.10/1.67 ! [C: $tType,B: $tType,A: $tType,P: C > $o,F4: A > B > C,Prod: product_prod @ A @ B] :
% 3.10/1.67 ( ( P @ ( product_case_prod @ A @ B @ C @ F4 @ Prod ) )
% 3.10/1.67 = ( ( Prod
% 3.10/1.67 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 3.10/1.67 => ( P @ ( F4 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_130_prod_Osplit__sel__asm,axiom,
% 3.10/1.67 ! [C: $tType,B: $tType,A: $tType,P: C > $o,F4: A > B > C,Prod: product_prod @ A @ B] :
% 3.10/1.67 ( ( P @ ( product_case_prod @ A @ B @ C @ F4 @ Prod ) )
% 3.10/1.67 = ( ~ ( ( Prod
% 3.10/1.67 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 3.10/1.67 & ~ ( P @ ( F4 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_131_prod_Oexhaust__sel,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 3.10/1.67 ( Prod
% 3.10/1.67 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_132_surjective__pairing,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 3.10/1.67 ( T2
% 3.10/1.67 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_133_prod__eqI,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,Q2: product_prod @ A @ B] :
% 3.10/1.67 ( ( ( product_fst @ A @ B @ P2 )
% 3.10/1.67 = ( product_fst @ A @ B @ Q2 ) )
% 3.10/1.67 => ( ( ( product_snd @ A @ B @ P2 )
% 3.10/1.67 = ( product_snd @ A @ B @ Q2 ) )
% 3.10/1.67 => ( P2 = Q2 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_134_prod_Oexpand,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
% 3.10/1.67 ( ( ( ( product_fst @ A @ B @ Prod )
% 3.10/1.67 = ( product_fst @ A @ B @ Prod2 ) )
% 3.10/1.67 & ( ( product_snd @ A @ B @ Prod )
% 3.10/1.67 = ( product_snd @ A @ B @ Prod2 ) ) )
% 3.10/1.67 => ( Prod = Prod2 ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_135_prod__eq__iff,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType] :
% 3.10/1.67 ( ( ^ [Y7: product_prod @ A @ B,Z3: product_prod @ A @ B] : ( Y7 = Z3 ) )
% 3.10/1.67 = ( ^ [S3: product_prod @ A @ B,T3: product_prod @ A @ B] :
% 3.10/1.67 ( ( ( product_fst @ A @ B @ S3 )
% 3.10/1.67 = ( product_fst @ A @ B @ T3 ) )
% 3.10/1.67 & ( ( product_snd @ A @ B @ S3 )
% 3.10/1.67 = ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_136_case__prod__beta,axiom,
% 3.10/1.67 ! [A: $tType,C: $tType,B: $tType] :
% 3.10/1.67 ( ( product_case_prod @ B @ C @ A )
% 3.10/1.67 = ( ^ [F3: B > C > A,P3: product_prod @ B @ C] : ( F3 @ ( product_fst @ B @ C @ P3 ) @ ( product_snd @ B @ C @ P3 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_137_prod_Ocase__eq__if,axiom,
% 3.10/1.67 ! [C: $tType,B: $tType,A: $tType] :
% 3.10/1.67 ( ( product_case_prod @ A @ B @ C )
% 3.10/1.67 = ( ^ [F3: A > B > C,Prod3: product_prod @ A @ B] : ( F3 @ ( product_fst @ A @ B @ Prod3 ) @ ( product_snd @ A @ B @ Prod3 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_138_Product__Type_OCollect__case__prodD,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,X: product_prod @ A @ B,A6: A > B > $o] :
% 3.10/1.67 ( ( member @ ( product_prod @ A @ B ) @ X @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A6 ) ) )
% 3.10/1.67 => ( A6 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_139_fst__conv,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,X1: A,X2: B] :
% 3.10/1.67 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X2 ) )
% 3.10/1.67 = X1 ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_140_fst__eqD,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,X: A,Y: B,A3: A] :
% 3.10/1.67 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
% 3.10/1.67 = A3 )
% 3.10/1.67 => ( X = A3 ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_141_snd__conv,axiom,
% 3.10/1.67 ! [Aa: $tType,A: $tType,X1: Aa,X2: A] :
% 3.10/1.67 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X2 ) )
% 3.10/1.67 = X2 ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_142_snd__eqD,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,X: B,Y: A,A3: A] :
% 3.10/1.67 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 3.10/1.67 = A3 )
% 3.10/1.67 => ( Y = A3 ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_143_prod_Oswap__def,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType] :
% 3.10/1.67 ( ( product_swap @ A @ B )
% 3.10/1.67 = ( ^ [P3: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P3 ) @ ( product_fst @ A @ B @ P3 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_144_exI__realizer,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,P: A > B > $o,Y: A,X: B] :
% 3.10/1.67 ( ( P @ Y @ X )
% 3.10/1.67 => ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_145_conjI__realizer,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,P: A > $o,P2: A,Q: B > $o,Q2: B] :
% 3.10/1.67 ( ( P @ P2 )
% 3.10/1.67 => ( ( Q @ Q2 )
% 3.10/1.67 => ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) )
% 3.10/1.67 & ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P2 @ Q2 ) ) ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_146_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,P: A > B > $o,X: A,Y: B,A3: product_prod @ A @ B] :
% 3.10/1.67 ( ( P @ X @ Y )
% 3.10/1.67 => ( ( A3
% 3.10/1.67 = ( product_Pair @ A @ B @ X @ Y ) )
% 3.10/1.67 => ( P @ ( product_fst @ A @ B @ A3 ) @ ( product_snd @ A @ B @ A3 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_147_sndI,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,X: product_prod @ A @ B,Y: A,Z2: B] :
% 3.10/1.67 ( ( X
% 3.10/1.67 = ( product_Pair @ A @ B @ Y @ Z2 ) )
% 3.10/1.67 => ( ( product_snd @ A @ B @ X )
% 3.10/1.67 = Z2 ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_148_fstI,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,X: product_prod @ A @ B,Y: A,Z2: B] :
% 3.10/1.67 ( ( X
% 3.10/1.67 = ( product_Pair @ A @ B @ Y @ Z2 ) )
% 3.10/1.67 => ( ( product_fst @ A @ B @ X )
% 3.10/1.67 = Y ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_149_eq__snd__iff,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,B3: A,P2: product_prod @ B @ A] :
% 3.10/1.67 ( ( B3
% 3.10/1.67 = ( product_snd @ B @ A @ P2 ) )
% 3.10/1.67 = ( ? [A2: B] :
% 3.10/1.67 ( P2
% 3.10/1.67 = ( product_Pair @ B @ A @ A2 @ B3 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_150_eq__fst__iff,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,A3: A,P2: product_prod @ A @ B] :
% 3.10/1.67 ( ( A3
% 3.10/1.67 = ( product_fst @ A @ B @ P2 ) )
% 3.10/1.67 = ( ? [B2: B] :
% 3.10/1.67 ( P2
% 3.10/1.67 = ( product_Pair @ A @ B @ A3 @ B2 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_151_Collect__split__mono__strong,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,X6: set @ A,A6: set @ ( product_prod @ A @ B ),Y8: set @ B,P: A > B > $o,Q: A > B > $o] :
% 3.10/1.67 ( ( X6
% 3.10/1.67 = ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A6 ) )
% 3.10/1.67 => ( ( Y8
% 3.10/1.67 = ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A6 ) )
% 3.10/1.67 => ( ! [X3: A] :
% 3.10/1.67 ( ( member @ A @ X3 @ X6 )
% 3.10/1.67 => ! [Xa: B] :
% 3.10/1.67 ( ( member @ B @ Xa @ Y8 )
% 3.10/1.67 => ( ( P @ X3 @ Xa )
% 3.10/1.67 => ( Q @ X3 @ Xa ) ) ) )
% 3.10/1.67 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) )
% 3.10/1.67 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q ) ) ) ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_152_subsetI,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 3.10/1.67 ( ! [X3: A] :
% 3.10/1.67 ( ( member @ A @ X3 @ A6 )
% 3.10/1.67 => ( member @ A @ X3 @ B6 ) )
% 3.10/1.67 => ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_153_subset__antisym,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 3.10/1.67 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 3.10/1.67 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A6 )
% 3.10/1.67 => ( A6 = B6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_154_Pow__iff,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 3.10/1.67 ( ( member @ ( set @ A ) @ A6 @ ( pow @ A @ B6 ) )
% 3.10/1.67 = ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_155_PowI,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 3.10/1.67 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 3.10/1.67 => ( member @ ( set @ A ) @ A6 @ ( pow @ A @ B6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_156_subrelI,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 3.10/1.67 ( ! [X3: A,Y3: B] :
% 3.10/1.67 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R )
% 3.10/1.67 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S ) )
% 3.10/1.67 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_157_Pow__mono,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 3.10/1.67 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 3.10/1.67 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow @ A @ A6 ) @ ( pow @ A @ B6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_158_top_Oextremum__uniqueI,axiom,
% 3.10/1.67 ! [A: $tType] :
% 3.10/1.67 ( ( order_top @ A @ ( type2 @ A ) )
% 3.10/1.67 => ! [A3: A] :
% 3.10/1.67 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 3.10/1.67 => ( A3
% 3.10/1.67 = ( top_top @ A ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_159_top_Oextremum__unique,axiom,
% 3.10/1.67 ! [A: $tType] :
% 3.10/1.67 ( ( order_top @ A @ ( type2 @ A ) )
% 3.10/1.67 => ! [A3: A] :
% 3.10/1.67 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 3.10/1.67 = ( A3
% 3.10/1.67 = ( top_top @ A ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_160_top__greatest,axiom,
% 3.10/1.67 ! [A: $tType] :
% 3.10/1.67 ( ( order_top @ A @ ( type2 @ A ) )
% 3.10/1.67 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_161_rev__finite__subset,axiom,
% 3.10/1.67 ! [A: $tType,B6: set @ A,A6: set @ A] :
% 3.10/1.67 ( ( finite_finite2 @ A @ B6 )
% 3.10/1.67 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 3.10/1.67 => ( finite_finite2 @ A @ A6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_162_infinite__super,axiom,
% 3.10/1.67 ! [A: $tType,S4: set @ A,T4: set @ A] :
% 3.10/1.67 ( ( ord_less_eq @ ( set @ A ) @ S4 @ T4 )
% 3.10/1.67 => ( ~ ( finite_finite2 @ A @ S4 )
% 3.10/1.67 => ~ ( finite_finite2 @ A @ T4 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_163_finite__subset,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 3.10/1.67 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 3.10/1.67 => ( ( finite_finite2 @ A @ B6 )
% 3.10/1.67 => ( finite_finite2 @ A @ A6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_164_subset__UNIV,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ A6 @ ( top_top @ ( set @ A ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_165_set__mp,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A,B6: set @ A,X: A] :
% 3.10/1.67 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 3.10/1.67 => ( ( member @ A @ X @ A6 )
% 3.10/1.67 => ( member @ A @ X @ B6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_166_in__mono,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A,B6: set @ A,X: A] :
% 3.10/1.67 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 3.10/1.67 => ( ( member @ A @ X @ A6 )
% 3.10/1.67 => ( member @ A @ X @ B6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_167_subsetD,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A,B6: set @ A,C2: A] :
% 3.10/1.67 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 3.10/1.67 => ( ( member @ A @ C2 @ A6 )
% 3.10/1.67 => ( member @ A @ C2 @ B6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_168_subsetCE,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A,B6: set @ A,C2: A] :
% 3.10/1.67 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 3.10/1.67 => ( ( member @ A @ C2 @ A6 )
% 3.10/1.67 => ( member @ A @ C2 @ B6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_169_equalityE,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 3.10/1.67 ( ( A6 = B6 )
% 3.10/1.67 => ~ ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 3.10/1.67 => ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_170_subset__eq,axiom,
% 3.10/1.67 ! [A: $tType] :
% 3.10/1.67 ( ( ord_less_eq @ ( set @ A ) )
% 3.10/1.67 = ( ^ [A7: set @ A,B7: set @ A] :
% 3.10/1.67 ! [X4: A] :
% 3.10/1.67 ( ( member @ A @ X4 @ A7 )
% 3.10/1.67 => ( member @ A @ X4 @ B7 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_171_equalityD1,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 3.10/1.67 ( ( A6 = B6 )
% 3.10/1.67 => ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_172_equalityD2,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 3.10/1.67 ( ( A6 = B6 )
% 3.10/1.67 => ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_173_set__rev__mp,axiom,
% 3.10/1.67 ! [A: $tType,X: A,A6: set @ A,B6: set @ A] :
% 3.10/1.67 ( ( member @ A @ X @ A6 )
% 3.10/1.67 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 3.10/1.67 => ( member @ A @ X @ B6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_174_subset__iff,axiom,
% 3.10/1.67 ! [A: $tType] :
% 3.10/1.67 ( ( ord_less_eq @ ( set @ A ) )
% 3.10/1.67 = ( ^ [A7: set @ A,B7: set @ A] :
% 3.10/1.67 ! [T3: A] :
% 3.10/1.67 ( ( member @ A @ T3 @ A7 )
% 3.10/1.67 => ( member @ A @ T3 @ B7 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_175_rev__subsetD,axiom,
% 3.10/1.67 ! [A: $tType,C2: A,A6: set @ A,B6: set @ A] :
% 3.10/1.67 ( ( member @ A @ C2 @ A6 )
% 3.10/1.67 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 3.10/1.67 => ( member @ A @ C2 @ B6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_176_subset__refl,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ A6 @ A6 ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_177_Collect__mono,axiom,
% 3.10/1.67 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.10/1.67 ( ! [X3: A] :
% 3.10/1.67 ( ( P @ X3 )
% 3.10/1.67 => ( Q @ X3 ) )
% 3.10/1.67 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_178_subset__trans,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A,B6: set @ A,C4: set @ A] :
% 3.10/1.67 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 3.10/1.67 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
% 3.10/1.67 => ( ord_less_eq @ ( set @ A ) @ A6 @ C4 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_179_set__eq__subset,axiom,
% 3.10/1.67 ! [A: $tType] :
% 3.10/1.67 ( ( ^ [Y7: set @ A,Z3: set @ A] : ( Y7 = Z3 ) )
% 3.10/1.67 = ( ^ [A7: set @ A,B7: set @ A] :
% 3.10/1.67 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 3.10/1.67 & ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_180_contra__subsetD,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A,B6: set @ A,C2: A] :
% 3.10/1.67 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 3.10/1.67 => ( ~ ( member @ A @ C2 @ B6 )
% 3.10/1.67 => ~ ( member @ A @ C2 @ A6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_181_Collect__mono__iff,axiom,
% 3.10/1.67 ! [A: $tType,P: A > $o,Q: A > $o] :
% 3.10/1.67 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 3.10/1.67 = ( ! [X4: A] :
% 3.10/1.67 ( ( P @ X4 )
% 3.10/1.67 => ( Q @ X4 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_182_PowD,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 3.10/1.67 ( ( member @ ( set @ A ) @ A6 @ ( pow @ A @ B6 ) )
% 3.10/1.67 => ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_183_image__mono,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,A6: set @ A,B6: set @ A,F4: A > B] :
% 3.10/1.67 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 3.10/1.67 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) @ ( image @ A @ B @ F4 @ B6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_184_image__subsetI,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,A6: set @ A,F4: A > B,B6: set @ B] :
% 3.10/1.67 ( ! [X3: A] :
% 3.10/1.67 ( ( member @ A @ X3 @ A6 )
% 3.10/1.67 => ( member @ B @ ( F4 @ X3 ) @ B6 ) )
% 3.10/1.67 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) @ B6 ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_185_subset__imageE,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,B6: set @ A,F4: B > A,A6: set @ B] :
% 3.10/1.67 ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F4 @ A6 ) )
% 3.10/1.67 => ~ ! [C5: set @ B] :
% 3.10/1.67 ( ( ord_less_eq @ ( set @ B ) @ C5 @ A6 )
% 3.10/1.67 => ( B6
% 3.10/1.67 != ( image @ B @ A @ F4 @ C5 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_186_image__subset__iff,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B,B6: set @ A] :
% 3.10/1.67 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F4 @ A6 ) @ B6 )
% 3.10/1.67 = ( ! [X4: B] :
% 3.10/1.67 ( ( member @ B @ X4 @ A6 )
% 3.10/1.67 => ( member @ A @ ( F4 @ X4 ) @ B6 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_187_subset__image__iff,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,B6: set @ A,F4: B > A,A6: set @ B] :
% 3.10/1.67 ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F4 @ A6 ) )
% 3.10/1.67 = ( ? [AA: set @ B] :
% 3.10/1.67 ( ( ord_less_eq @ ( set @ B ) @ AA @ A6 )
% 3.10/1.67 & ( B6
% 3.10/1.67 = ( image @ B @ A @ F4 @ AA ) ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_188_image__Pow__mono,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,F4: B > A,A6: set @ B,B6: set @ A] :
% 3.10/1.67 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F4 @ A6 ) @ B6 )
% 3.10/1.67 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F4 ) @ ( pow @ B @ A6 ) ) @ ( pow @ A @ B6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_189_finite__surj,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,A6: set @ A,B6: set @ B,F4: A > B] :
% 3.10/1.67 ( ( finite_finite2 @ A @ A6 )
% 3.10/1.67 => ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( image @ A @ B @ F4 @ A6 ) )
% 3.10/1.67 => ( finite_finite2 @ B @ B6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_190_finite__subset__image,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,B6: set @ A,F4: B > A,A6: set @ B] :
% 3.10/1.67 ( ( finite_finite2 @ A @ B6 )
% 3.10/1.67 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F4 @ A6 ) )
% 3.10/1.67 => ? [C5: set @ B] :
% 3.10/1.67 ( ( ord_less_eq @ ( set @ B ) @ C5 @ A6 )
% 3.10/1.67 & ( finite_finite2 @ B @ C5 )
% 3.10/1.67 & ( B6
% 3.10/1.67 = ( image @ B @ A @ F4 @ C5 ) ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_191_relChain__def,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType] :
% 3.10/1.67 ( ( ord @ B @ ( type2 @ B ) )
% 3.10/1.67 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 3.10/1.67 = ( ^ [R2: set @ ( product_prod @ A @ A ),As4: A > B] :
% 3.10/1.67 ! [I: A,J: A] :
% 3.10/1.67 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R2 )
% 3.10/1.67 => ( ord_less_eq @ B @ ( As4 @ I ) @ ( As4 @ J ) ) ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_192_fstOp__def,axiom,
% 3.10/1.67 ! [C: $tType,B: $tType,A: $tType] :
% 3.10/1.67 ( ( bNF_fstOp @ A @ B @ C )
% 3.10/1.67 = ( ^ [P4: A > B > $o,Q3: B > C > $o,Ac: product_prod @ A @ C] : ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Ac ) @ ( bNF_pick_middlep @ A @ B @ C @ P4 @ Q3 @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_193_sndOp__def,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,C: $tType] :
% 3.10/1.67 ( ( bNF_sndOp @ C @ A @ B )
% 3.10/1.67 = ( ^ [P4: C > A > $o,Q3: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P4 @ Q3 @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_194_accp__subset,axiom,
% 3.10/1.67 ! [A: $tType,R1: A > A > $o,R22: A > A > $o] :
% 3.10/1.67 ( ( ord_less_eq @ ( A > A > $o ) @ R1 @ R22 )
% 3.10/1.67 => ( ord_less_eq @ ( A > $o ) @ ( accp @ A @ R22 ) @ ( accp @ A @ R1 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_195_accp__subset__induct,axiom,
% 3.10/1.67 ! [A: $tType,D3: A > $o,R3: A > A > $o,X: A,P: A > $o] :
% 3.10/1.67 ( ( ord_less_eq @ ( A > $o ) @ D3 @ ( accp @ A @ R3 ) )
% 3.10/1.67 => ( ! [X3: A,Z: A] :
% 3.10/1.67 ( ( D3 @ X3 )
% 3.10/1.67 => ( ( R3 @ Z @ X3 )
% 3.10/1.67 => ( D3 @ Z ) ) )
% 3.10/1.67 => ( ( D3 @ X )
% 3.10/1.67 => ( ! [X3: A] :
% 3.10/1.67 ( ( D3 @ X3 )
% 3.10/1.67 => ( ! [Z4: A] :
% 3.10/1.67 ( ( R3 @ Z4 @ X3 )
% 3.10/1.67 => ( P @ Z4 ) )
% 3.10/1.67 => ( P @ X3 ) ) )
% 3.10/1.67 => ( P @ X ) ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_196_Collect__case__prod__mono,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,A6: A > B > $o,B6: A > B > $o] :
% 3.10/1.67 ( ( ord_less_eq @ ( A > B > $o ) @ A6 @ B6 )
% 3.10/1.67 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A6 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B6 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_197_finite__vimageD_H,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,F4: A > B,A6: set @ B] :
% 3.10/1.67 ( ( finite_finite2 @ A @ ( vimage @ A @ B @ F4 @ A6 ) )
% 3.10/1.67 => ( ( ord_less_eq @ ( set @ B ) @ A6 @ ( image @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) ) )
% 3.10/1.67 => ( finite_finite2 @ B @ A6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_198_image2__eqI,axiom,
% 3.10/1.67 ! [A: $tType,C: $tType,B: $tType,B3: A,F4: B > A,X: B,C2: C,G3: B > C,A6: set @ B] :
% 3.10/1.67 ( ( B3
% 3.10/1.67 = ( F4 @ X ) )
% 3.10/1.67 => ( ( C2
% 3.10/1.67 = ( G3 @ X ) )
% 3.10/1.67 => ( ( member @ B @ X @ A6 )
% 3.10/1.67 => ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B3 @ C2 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A6 @ F4 @ G3 ) ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_199_vimageI,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,F4: B > A,A3: B,B3: A,B6: set @ A] :
% 3.10/1.67 ( ( ( F4 @ A3 )
% 3.10/1.67 = B3 )
% 3.10/1.67 => ( ( member @ A @ B3 @ B6 )
% 3.10/1.67 => ( member @ B @ A3 @ ( vimage @ B @ A @ F4 @ B6 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_200_vimage__eq,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,A3: A,F4: A > B,B6: set @ B] :
% 3.10/1.67 ( ( member @ A @ A3 @ ( vimage @ A @ B @ F4 @ B6 ) )
% 3.10/1.67 = ( member @ B @ ( F4 @ A3 ) @ B6 ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_201_vimage__UNIV,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,F4: A > B] :
% 3.10/1.67 ( ( vimage @ A @ B @ F4 @ ( top_top @ ( set @ B ) ) )
% 3.10/1.67 = ( top_top @ ( set @ A ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_202_vimage__mono,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,A6: set @ A,B6: set @ A,F4: B > A] :
% 3.10/1.67 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 3.10/1.67 => ( ord_less_eq @ ( set @ B ) @ ( vimage @ B @ A @ F4 @ A6 ) @ ( vimage @ B @ A @ F4 @ B6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_203_image__subset__iff__subset__vimage,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,F4: B > A,A6: set @ B,B6: set @ A] :
% 3.10/1.67 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F4 @ A6 ) @ B6 )
% 3.10/1.67 = ( ord_less_eq @ ( set @ B ) @ A6 @ ( vimage @ B @ A @ F4 @ B6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_204_image__vimage__subset,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,F4: B > A,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F4 @ ( vimage @ B @ A @ F4 @ A6 ) ) @ A6 ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_205_surj__image__vimage__eq,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,F4: B > A,A6: set @ A] :
% 3.10/1.67 ( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 3.10/1.67 = ( top_top @ ( set @ A ) ) )
% 3.10/1.67 => ( ( image @ B @ A @ F4 @ ( vimage @ B @ A @ F4 @ A6 ) )
% 3.10/1.67 = A6 ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_206_vimageD,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,A3: A,F4: A > B,A6: set @ B] :
% 3.10/1.67 ( ( member @ A @ A3 @ ( vimage @ A @ B @ F4 @ A6 ) )
% 3.10/1.67 => ( member @ B @ ( F4 @ A3 ) @ A6 ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_207_vimageE,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,A3: A,F4: A > B,B6: set @ B] :
% 3.10/1.67 ( ( member @ A @ A3 @ ( vimage @ A @ B @ F4 @ B6 ) )
% 3.10/1.67 => ( member @ B @ ( F4 @ A3 ) @ B6 ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_208_vimageI2,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,F4: B > A,A3: B,A6: set @ A] :
% 3.10/1.67 ( ( member @ A @ ( F4 @ A3 ) @ A6 )
% 3.10/1.67 => ( member @ B @ A3 @ ( vimage @ B @ A @ F4 @ A6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_209_vimage__Collect,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,P: B > $o,F4: A > B,Q: A > $o] :
% 3.10/1.67 ( ! [X3: A] :
% 3.10/1.67 ( ( P @ ( F4 @ X3 ) )
% 3.10/1.67 = ( Q @ X3 ) )
% 3.10/1.67 => ( ( vimage @ A @ B @ F4 @ ( collect @ B @ P ) )
% 3.10/1.67 = ( collect @ A @ Q ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_210_finite__vimageD,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,H: A > B,F5: set @ B] :
% 3.10/1.67 ( ( finite_finite2 @ A @ ( vimage @ A @ B @ H @ F5 ) )
% 3.10/1.67 => ( ( ( image @ A @ B @ H @ ( top_top @ ( set @ A ) ) )
% 3.10/1.67 = ( top_top @ ( set @ B ) ) )
% 3.10/1.67 => ( finite_finite2 @ B @ F5 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_211_vimage__subsetD,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,F4: B > A,B6: set @ A,A6: set @ B] :
% 3.10/1.67 ( ( ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) )
% 3.10/1.67 = ( top_top @ ( set @ A ) ) )
% 3.10/1.67 => ( ( ord_less_eq @ ( set @ B ) @ ( vimage @ B @ A @ F4 @ B6 ) @ A6 )
% 3.10/1.67 => ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F4 @ A6 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_212_image__vimage__eq,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,F4: B > A,A6: set @ A] :
% 3.10/1.67 ( ( image @ B @ A @ F4 @ ( vimage @ B @ A @ F4 @ A6 ) )
% 3.10/1.67 = ( inf_inf @ ( set @ A ) @ A6 @ ( image @ B @ A @ F4 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_213_vimage__subsetI,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,F4: A > B,B6: set @ B,A6: set @ A] :
% 3.10/1.67 ( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 3.10/1.67 => ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( image @ A @ B @ F4 @ A6 ) )
% 3.10/1.67 => ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F4 @ B6 ) @ A6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_214_Int__iff,axiom,
% 3.10/1.67 ! [A: $tType,C2: A,A6: set @ A,B6: set @ A] :
% 3.10/1.67 ( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) )
% 3.10/1.67 = ( ( member @ A @ C2 @ A6 )
% 3.10/1.67 & ( member @ A @ C2 @ B6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_215_IntI,axiom,
% 3.10/1.67 ! [A: $tType,C2: A,A6: set @ A,B6: set @ A] :
% 3.10/1.67 ( ( member @ A @ C2 @ A6 )
% 3.10/1.67 => ( ( member @ A @ C2 @ B6 )
% 3.10/1.67 => ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_216_finite__Int,axiom,
% 3.10/1.67 ! [A: $tType,F5: set @ A,G5: set @ A] :
% 3.10/1.67 ( ( ( finite_finite2 @ A @ F5 )
% 3.10/1.67 | ( finite_finite2 @ A @ G5 ) )
% 3.10/1.67 => ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ F5 @ G5 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_217_Int__subset__iff,axiom,
% 3.10/1.67 ! [A: $tType,C4: set @ A,A6: set @ A,B6: set @ A] :
% 3.10/1.67 ( ( ord_less_eq @ ( set @ A ) @ C4 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) )
% 3.10/1.67 = ( ( ord_less_eq @ ( set @ A ) @ C4 @ A6 )
% 3.10/1.67 & ( ord_less_eq @ ( set @ A ) @ C4 @ B6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_218_Int__UNIV,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 3.10/1.67 ( ( ( inf_inf @ ( set @ A ) @ A6 @ B6 )
% 3.10/1.67 = ( top_top @ ( set @ A ) ) )
% 3.10/1.67 = ( ( A6
% 3.10/1.67 = ( top_top @ ( set @ A ) ) )
% 3.10/1.67 & ( B6
% 3.10/1.67 = ( top_top @ ( set @ A ) ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_219_vimage__Int,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,F4: A > B,A6: set @ B,B6: set @ B] :
% 3.10/1.67 ( ( vimage @ A @ B @ F4 @ ( inf_inf @ ( set @ B ) @ A6 @ B6 ) )
% 3.10/1.67 = ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F4 @ A6 ) @ ( vimage @ A @ B @ F4 @ B6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_220_Pow__Int__eq,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 3.10/1.67 ( ( pow @ A @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) )
% 3.10/1.67 = ( inf_inf @ ( set @ ( set @ A ) ) @ ( pow @ A @ A6 ) @ ( pow @ A @ B6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_221_Int__mono,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A,C4: set @ A,B6: set @ A,D3: set @ A] :
% 3.10/1.67 ( ( ord_less_eq @ ( set @ A ) @ A6 @ C4 )
% 3.10/1.67 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ D3 )
% 3.10/1.67 => ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) @ ( inf_inf @ ( set @ A ) @ C4 @ D3 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_222_Int__lower1,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A,B6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) @ A6 ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_223_Int__lower2,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A,B6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) @ B6 ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_224_Int__absorb1,axiom,
% 3.10/1.67 ! [A: $tType,B6: set @ A,A6: set @ A] :
% 3.10/1.67 ( ( ord_less_eq @ ( set @ A ) @ B6 @ A6 )
% 3.10/1.67 => ( ( inf_inf @ ( set @ A ) @ A6 @ B6 )
% 3.10/1.67 = B6 ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_225_Int__absorb2,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 3.10/1.67 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 3.10/1.67 => ( ( inf_inf @ ( set @ A ) @ A6 @ B6 )
% 3.10/1.67 = A6 ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_226_Int__greatest,axiom,
% 3.10/1.67 ! [A: $tType,C4: set @ A,A6: set @ A,B6: set @ A] :
% 3.10/1.67 ( ( ord_less_eq @ ( set @ A ) @ C4 @ A6 )
% 3.10/1.67 => ( ( ord_less_eq @ ( set @ A ) @ C4 @ B6 )
% 3.10/1.67 => ( ord_less_eq @ ( set @ A ) @ C4 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_227_Int__Collect__mono,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A,B6: set @ A,P: A > $o,Q: A > $o] :
% 3.10/1.67 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 3.10/1.67 => ( ! [X3: A] :
% 3.10/1.67 ( ( member @ A @ X3 @ A6 )
% 3.10/1.67 => ( ( P @ X3 )
% 3.10/1.67 => ( Q @ X3 ) ) )
% 3.10/1.67 => ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ ( collect @ A @ P ) ) @ ( inf_inf @ ( set @ A ) @ B6 @ ( collect @ A @ Q ) ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_228_inj__on__image__Int,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,F4: A > B,C4: set @ A,A6: set @ A,B6: set @ A] :
% 3.10/1.67 ( ( inj_on @ A @ B @ F4 @ C4 )
% 3.10/1.67 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ C4 )
% 3.10/1.67 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
% 3.10/1.67 => ( ( image @ A @ B @ F4 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) )
% 3.10/1.67 = ( inf_inf @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) @ ( image @ A @ B @ F4 @ B6 ) ) ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_229_inj__on__image__eq__iff,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,F4: A > B,C4: set @ A,A6: set @ A,B6: set @ A] :
% 3.10/1.67 ( ( inj_on @ A @ B @ F4 @ C4 )
% 3.10/1.67 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ C4 )
% 3.10/1.67 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
% 3.10/1.67 => ( ( ( image @ A @ B @ F4 @ A6 )
% 3.10/1.67 = ( image @ A @ B @ F4 @ B6 ) )
% 3.10/1.67 = ( A6 = B6 ) ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_230_inj__on__image__mem__iff,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,F4: A > B,B6: set @ A,A3: A,A6: set @ A] :
% 3.10/1.67 ( ( inj_on @ A @ B @ F4 @ B6 )
% 3.10/1.67 => ( ( member @ A @ A3 @ B6 )
% 3.10/1.67 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 3.10/1.67 => ( ( member @ B @ ( F4 @ A3 ) @ ( image @ A @ B @ F4 @ A6 ) )
% 3.10/1.67 = ( member @ A @ A3 @ A6 ) ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_231_inj__on__image__mem__iff__alt,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,F4: A > B,B6: set @ A,A6: set @ A,A3: A] :
% 3.10/1.67 ( ( inj_on @ A @ B @ F4 @ B6 )
% 3.10/1.67 => ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
% 3.10/1.67 => ( ( member @ B @ ( F4 @ A3 ) @ ( image @ A @ B @ F4 @ A6 ) )
% 3.10/1.67 => ( ( member @ A @ A3 @ B6 )
% 3.10/1.67 => ( member @ A @ A3 @ A6 ) ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_232_image__Int__subset,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B,B6: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F4 @ ( inf_inf @ ( set @ B ) @ A6 @ B6 ) ) @ ( inf_inf @ ( set @ A ) @ ( image @ B @ A @ F4 @ A6 ) @ ( image @ B @ A @ F4 @ B6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_233_vimage__inter__cong,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,S4: set @ A,F4: A > B,G3: A > B,Y: set @ B] :
% 3.10/1.67 ( ! [W: A] :
% 3.10/1.67 ( ( member @ A @ W @ S4 )
% 3.10/1.67 => ( ( F4 @ W )
% 3.10/1.67 = ( G3 @ W ) ) )
% 3.10/1.67 => ( ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F4 @ Y ) @ S4 )
% 3.10/1.67 = ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ G3 @ Y ) @ S4 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_234_finite__vimage__IntI,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,F5: set @ A,H: B > A,A6: set @ B] :
% 3.10/1.67 ( ( finite_finite2 @ A @ F5 )
% 3.10/1.67 => ( ( inj_on @ B @ A @ H @ A6 )
% 3.10/1.67 => ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H @ F5 ) @ A6 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_235_image__Int,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,F4: A > B,A6: set @ A,B6: set @ A] :
% 3.10/1.67 ( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 3.10/1.67 => ( ( image @ A @ B @ F4 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) )
% 3.10/1.67 = ( inf_inf @ ( set @ B ) @ ( image @ A @ B @ F4 @ A6 ) @ ( image @ A @ B @ F4 @ B6 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_236_inj__on__image__iff,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,A6: set @ A,G3: A > B,F4: A > A] :
% 3.10/1.67 ( ! [X3: A] :
% 3.10/1.67 ( ( member @ A @ X3 @ A6 )
% 3.10/1.67 => ! [Xa: A] :
% 3.10/1.67 ( ( member @ A @ Xa @ A6 )
% 3.10/1.67 => ( ( ( G3 @ ( F4 @ X3 ) )
% 3.10/1.67 = ( G3 @ ( F4 @ Xa ) ) )
% 3.10/1.67 = ( ( G3 @ X3 )
% 3.10/1.67 = ( G3 @ Xa ) ) ) ) )
% 3.10/1.67 => ( ( inj_on @ A @ A @ F4 @ A6 )
% 3.10/1.67 => ( ( inj_on @ A @ B @ G3 @ ( image @ A @ A @ F4 @ A6 ) )
% 3.10/1.67 = ( inj_on @ A @ B @ G3 @ A6 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_237_inj__swap,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,A6: set @ ( product_prod @ A @ B )] : ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) @ ( product_swap @ A @ B ) @ A6 ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_238_Int__left__commute,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A,B6: set @ A,C4: set @ A] :
% 3.10/1.67 ( ( inf_inf @ ( set @ A ) @ A6 @ ( inf_inf @ ( set @ A ) @ B6 @ C4 ) )
% 3.10/1.67 = ( inf_inf @ ( set @ A ) @ B6 @ ( inf_inf @ ( set @ A ) @ A6 @ C4 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_239_Int__left__absorb,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A,B6: set @ A] :
% 3.10/1.67 ( ( inf_inf @ ( set @ A ) @ A6 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) )
% 3.10/1.67 = ( inf_inf @ ( set @ A ) @ A6 @ B6 ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_240_Int__commute,axiom,
% 3.10/1.67 ! [A: $tType] :
% 3.10/1.67 ( ( inf_inf @ ( set @ A ) )
% 3.10/1.67 = ( ^ [A7: set @ A,B7: set @ A] : ( inf_inf @ ( set @ A ) @ B7 @ A7 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_241_Int__absorb,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A] :
% 3.10/1.67 ( ( inf_inf @ ( set @ A ) @ A6 @ A6 )
% 3.10/1.67 = A6 ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_242_Int__assoc,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A,B6: set @ A,C4: set @ A] :
% 3.10/1.67 ( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) @ C4 )
% 3.10/1.67 = ( inf_inf @ ( set @ A ) @ A6 @ ( inf_inf @ ( set @ A ) @ B6 @ C4 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_243_IntD2,axiom,
% 3.10/1.67 ! [A: $tType,C2: A,A6: set @ A,B6: set @ A] :
% 3.10/1.67 ( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) )
% 3.10/1.67 => ( member @ A @ C2 @ B6 ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_244_IntD1,axiom,
% 3.10/1.67 ! [A: $tType,C2: A,A6: set @ A,B6: set @ A] :
% 3.10/1.67 ( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) )
% 3.10/1.67 => ( member @ A @ C2 @ A6 ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_245_IntE,axiom,
% 3.10/1.67 ! [A: $tType,C2: A,A6: set @ A,B6: set @ A] :
% 3.10/1.67 ( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) )
% 3.10/1.67 => ~ ( ( member @ A @ C2 @ A6 )
% 3.10/1.67 => ~ ( member @ A @ C2 @ B6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_246_Int__UNIV__left,axiom,
% 3.10/1.67 ! [A: $tType,B6: set @ A] :
% 3.10/1.67 ( ( inf_inf @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B6 )
% 3.10/1.67 = B6 ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_247_Int__UNIV__right,axiom,
% 3.10/1.67 ! [A: $tType,A6: set @ A] :
% 3.10/1.67 ( ( inf_inf @ ( set @ A ) @ A6 @ ( top_top @ ( set @ A ) ) )
% 3.10/1.67 = A6 ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_248_inj__eq,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,F4: A > B,X: A,Y: A] :
% 3.10/1.67 ( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 3.10/1.67 => ( ( ( F4 @ X )
% 3.10/1.67 = ( F4 @ Y ) )
% 3.10/1.67 = ( X = Y ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_249_injI,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,F4: A > B] :
% 3.10/1.67 ( ! [X3: A,Y3: A] :
% 3.10/1.67 ( ( ( F4 @ X3 )
% 3.10/1.67 = ( F4 @ Y3 ) )
% 3.10/1.67 => ( X3 = Y3 ) )
% 3.10/1.67 => ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_250_injD,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,F4: A > B,X: A,Y: A] :
% 3.10/1.67 ( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 3.10/1.67 => ( ( ( F4 @ X )
% 3.10/1.67 = ( F4 @ Y ) )
% 3.10/1.67 => ( X = Y ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_251_range__ex1__eq,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,F4: A > B,B3: B] :
% 3.10/1.67 ( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 3.10/1.67 => ( ( member @ B @ B3 @ ( image @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) ) )
% 3.10/1.67 = ( ? [X4: A] :
% 3.10/1.67 ( ( B3
% 3.10/1.67 = ( F4 @ X4 ) )
% 3.10/1.67 & ! [Y5: A] :
% 3.10/1.67 ( ( B3
% 3.10/1.67 = ( F4 @ Y5 ) )
% 3.10/1.67 => ( Y5 = X4 ) ) ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_252_inj__image__eq__iff,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,F4: A > B,A6: set @ A,B6: set @ A] :
% 3.10/1.67 ( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 3.10/1.67 => ( ( ( image @ A @ B @ F4 @ A6 )
% 3.10/1.67 = ( image @ A @ B @ F4 @ B6 ) )
% 3.10/1.67 = ( A6 = B6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_253_inj__image__mem__iff,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,F4: A > B,A3: A,A6: set @ A] :
% 3.10/1.67 ( ( inj_on @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) )
% 3.10/1.67 => ( ( member @ B @ ( F4 @ A3 ) @ ( image @ A @ B @ F4 @ A6 ) )
% 3.10/1.67 = ( member @ A @ A3 @ A6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_254_finite__imageD,axiom,
% 3.10/1.67 ! [A: $tType,B: $tType,F4: B > A,A6: set @ B] :
% 3.10/1.67 ( ( finite_finite2 @ A @ ( image @ B @ A @ F4 @ A6 ) )
% 3.10/1.67 => ( ( inj_on @ B @ A @ F4 @ A6 )
% 3.10/1.67 => ( finite_finite2 @ B @ A6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(fact_255_finite__image__iff,axiom,
% 3.10/1.67 ! [B: $tType,A: $tType,F4: A > B,A6: set @ A] :
% 3.10/1.67 ( ( inj_on @ A @ B @ F4 @ A6 )
% 3.10/1.67 => ( ( finite_finite2 @ B @ ( image @ A @ B @ F4 @ A6 ) )
% 3.10/1.67 = ( finite_finite2 @ A @ A6 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 3.10/1.67 ! [A8: $tType,A9: $tType] :
% 3.10/1.67 ( ( order_top @ A9 @ ( type2 @ A9 ) )
% 3.10/1.67 => ( order_top @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 3.10/1.67 ! [A8: $tType,A9: $tType] :
% 3.10/1.67 ( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 3.10/1.67 & ( finite_finite @ A9 @ ( type2 @ A9 ) ) )
% 3.10/1.67 => ( finite_finite @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(tcon_fun___Orderings_Otop,axiom,
% 3.10/1.67 ! [A8: $tType,A9: $tType] :
% 3.10/1.67 ( ( top @ A9 @ ( type2 @ A9 ) )
% 3.10/1.67 => ( top @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(tcon_fun___Orderings_Oord,axiom,
% 3.10/1.67 ! [A8: $tType,A9: $tType] :
% 3.10/1.67 ( ( ord @ A9 @ ( type2 @ A9 ) )
% 3.10/1.67 => ( ord @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(tcon_Set_Oset___Orderings_Oorder__top_1,axiom,
% 3.10/1.67 ! [A8: $tType] : ( order_top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(tcon_Set_Oset___Finite__Set_Ofinite_2,axiom,
% 3.10/1.67 ! [A8: $tType] :
% 3.10/1.67 ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 3.10/1.67 => ( finite_finite @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(tcon_Set_Oset___Orderings_Otop_3,axiom,
% 3.10/1.67 ! [A8: $tType] : ( top @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(tcon_Set_Oset___Orderings_Oord_4,axiom,
% 3.10/1.67 ! [A8: $tType] : ( ord @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(tcon_HOL_Obool___Orderings_Oorder__top_5,axiom,
% 3.10/1.67 order_top @ $o @ ( type2 @ $o ) ).
% 3.10/1.67
% 3.10/1.67 thf(tcon_HOL_Obool___Finite__Set_Ofinite_6,axiom,
% 3.10/1.67 finite_finite @ $o @ ( type2 @ $o ) ).
% 3.10/1.67
% 3.10/1.67 thf(tcon_HOL_Obool___Orderings_Otop_7,axiom,
% 3.10/1.67 top @ $o @ ( type2 @ $o ) ).
% 3.10/1.67
% 3.10/1.67 thf(tcon_HOL_Obool___Orderings_Oord_8,axiom,
% 3.10/1.67 ord @ $o @ ( type2 @ $o ) ).
% 3.10/1.67
% 3.10/1.67 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_9,axiom,
% 3.10/1.67 ! [A8: $tType,A9: $tType] :
% 3.10/1.67 ( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 3.10/1.67 & ( finite_finite @ A9 @ ( type2 @ A9 ) ) )
% 3.10/1.67 => ( finite_finite @ ( sum_sum @ A8 @ A9 ) @ ( type2 @ ( sum_sum @ A8 @ A9 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(tcon_Option_Ooption___Finite__Set_Ofinite_10,axiom,
% 3.10/1.67 ! [A8: $tType] :
% 3.10/1.67 ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 3.10/1.67 => ( finite_finite @ ( option @ A8 ) @ ( type2 @ ( option @ A8 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_11,axiom,
% 3.10/1.67 ! [A8: $tType,A9: $tType] :
% 3.10/1.67 ( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
% 3.10/1.67 & ( finite_finite @ A9 @ ( type2 @ A9 ) ) )
% 3.10/1.67 => ( finite_finite @ ( product_prod @ A8 @ A9 ) @ ( type2 @ ( product_prod @ A8 @ A9 ) ) ) ) ).
% 3.10/1.67
% 3.10/1.67 thf(conj_0,conjecture,
% 3.10/1.67 ( ( root @ ( unfold @ ( product_prod @ dtree @ dtree ) @ parall1914194362_par_r @ parall1914194347_par_c @ ( product_Pair @ dtree @ dtree @ tr1 @ tr2 ) ) )
% 3.10/1.67 = ( parall1518086719_Nplus @ ( root @ tr1 ) @ ( root @ tr2 ) ) ) ).
% 3.10/1.67 ------------------------
% 3.16/1.70 % (20912)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_20809 for (2999ds/2Mi)
% 3.16/1.70 % (20908)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_20809 for (2999ds/183Mi)
% 3.16/1.70 % (20911)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_20809 for (2999ds/2Mi)
% 3.16/1.70 % (20914)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_20809 for (2999ds/18Mi)
% 3.16/1.70 % (20909)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_20809 for (2999ds/4Mi)
% 3.16/1.70 % (20913)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_20809 for (2999ds/275Mi)
% 3.16/1.70 % (20910)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_20809 for (2999ds/27Mi)
% 3.16/1.70 % (20911)Instruction limit reached!
% 3.16/1.70 % (20911)------------------------------
% 3.16/1.70 % (20911)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.16/1.70 % (20911)Termination reason: Unknown
% 3.16/1.70 % (20911)Termination phase: shuffling
% 3.16/1.70 % (20912)Instruction limit reached!
% 3.16/1.70 % (20912)------------------------------
% 3.16/1.70 % (20912)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.16/1.70 % (20912)Termination reason: Unknown
% 3.16/1.70 % (20912)Termination phase: shuffling
% 3.16/1.70
% 3.16/1.70 % (20912)Memory used [KB]: 1407
% 3.16/1.70 % (20912)Time elapsed: 0.003 s
% 3.16/1.70 % (20912)Instructions burned: 3 (million)
% 3.16/1.70 % (20912)------------------------------
% 3.16/1.70 % (20912)------------------------------
% 3.16/1.70
% 3.16/1.70 % (20911)Memory used [KB]: 1407
% 3.16/1.70 % (20911)Time elapsed: 0.003 s
% 3.16/1.70 % (20911)Instructions burned: 2 (million)
% 3.16/1.70 % (20911)------------------------------
% 3.16/1.70 % (20911)------------------------------
% 3.16/1.70 % (20909)Instruction limit reached!
% 3.16/1.70 % (20909)------------------------------
% 3.16/1.70 % (20909)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.16/1.70 % (20909)Termination reason: Unknown
% 3.16/1.70 % (20909)Termination phase: shuffling
% 3.16/1.70
% 3.16/1.70 % (20909)Memory used [KB]: 1407
% 3.16/1.70 % (20909)Time elapsed: 0.004 s
% 3.16/1.70 % (20909)Instructions burned: 4 (million)
% 3.16/1.70 % (20909)------------------------------
% 3.16/1.70 % (20909)------------------------------
% 3.16/1.71 % (20914)Instruction limit reached!
% 3.16/1.71 % (20914)------------------------------
% 3.16/1.71 % (20914)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.16/1.71 % (20914)Termination reason: Unknown
% 3.16/1.71 % (20914)Termination phase: shuffling
% 3.16/1.71
% 3.16/1.71 % (20914)Memory used [KB]: 1791
% 3.16/1.71 % (20914)Time elapsed: 0.011 s
% 3.16/1.71 % (20914)Instructions burned: 18 (million)
% 3.16/1.71 % (20914)------------------------------
% 3.16/1.71 % (20914)------------------------------
% 3.16/1.72 % (20910)Instruction limit reached!
% 3.16/1.72 % (20910)------------------------------
% 3.16/1.72 % (20910)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.16/1.72 % (20910)Termination reason: Unknown
% 3.16/1.72 % (20910)Termination phase: Property scanning
% 3.16/1.72
% 3.16/1.72 % (20910)Memory used [KB]: 1918
% 3.16/1.72 % (20910)Time elapsed: 0.016 s
% 3.16/1.72 % (20910)Instructions burned: 28 (million)
% 3.16/1.72 % (20910)------------------------------
% 3.16/1.72 % (20910)------------------------------
% 3.16/1.72 % (20915)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_20809 for (2999ds/3Mi)
% 3.16/1.72 % (20916)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_20809 for (2999ds/37Mi)
% 3.16/1.72 % (20917)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_20809 for (2999ds/15Mi)
% 3.16/1.72 % (20915)Instruction limit reached!
% 3.16/1.72 % (20915)------------------------------
% 3.16/1.72 % (20915)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.16/1.72 % (20915)Termination reason: Unknown
% 3.16/1.72 % (20915)Termination phase: shuffling
% 3.16/1.72
% 3.16/1.72 % (20915)Memory used [KB]: 1407
% 3.16/1.72 % (20915)Time elapsed: 0.003 s
% 3.16/1.72 % (20915)Instructions burned: 3 (million)
% 3.16/1.72 % (20915)------------------------------
% 3.16/1.72 % (20915)------------------------------
% 3.16/1.73 % (20918)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_20809 for (2999ds/3Mi)
% 3.16/1.73 % (20917)Instruction limit reached!
% 3.16/1.73 % (20917)------------------------------
% 3.16/1.73 % (20917)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.16/1.73 % (20917)Termination reason: Unknown
% 3.16/1.73 % (20917)Termination phase: shuffling
% 3.16/1.73
% 3.16/1.73 % (20917)Memory used [KB]: 1663
% 3.16/1.73 % (20917)Time elapsed: 0.009 s
% 3.16/1.73 % (20917)Instructions burned: 15 (million)
% 3.16/1.73 % (20917)------------------------------
% 3.16/1.73 % (20917)------------------------------
% 3.16/1.73 % (20918)Instruction limit reached!
% 3.16/1.73 % (20918)------------------------------
% 3.16/1.73 % (20918)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.16/1.73 % (20918)Termination reason: Unknown
% 3.16/1.73 % (20918)Termination phase: shuffling
% 3.16/1.73
% 3.16/1.73 % (20918)Memory used [KB]: 1407
% 3.16/1.73 % (20918)Time elapsed: 0.003 s
% 3.16/1.73 % (20918)Instructions burned: 3 (million)
% 3.16/1.73 % (20918)------------------------------
% 3.16/1.73 % (20918)------------------------------
% 3.16/1.73 % (20919)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_20809 for (2999ds/1041Mi)
% 3.16/1.73 % (20913)First to succeed.
% 3.16/1.74 % (20913)Refutation found. Thanks to Tanya!
% 3.16/1.74 % SZS status Theorem for DTF2THF_20809
% 3.16/1.74 % SZS output start Proof for DTF2THF_20809
% 3.16/1.74 thf(type_def_5, type, itself: ($tType) > $tType).
% 3.16/1.74 thf(type_def_7, type, product_prod: ($tType * $tType) > $tType).
% 3.16/1.74 thf(type_def_8, type, set: ($tType) > $tType).
% 3.16/1.74 thf(type_def_9, type, n: $tType).
% 3.16/1.74 thf(type_def_10, type, t: $tType).
% 3.16/1.74 thf(type_def_11, type, dtree: $tType).
% 3.16/1.74 thf(type_def_12, type, sum_sum: ($tType * $tType) > $tType).
% 3.16/1.74 thf(type_def_13, type, option: ($tType) > $tType).
% 3.16/1.74 thf(func_def_0, type, product_prod: ($tType * $tType) > $tType).
% 3.16/1.74 thf(func_def_1, type, sum_sum: ($tType * $tType) > $tType).
% 3.16/1.74 thf(func_def_2, type, option: ($tType) > $tType).
% 3.16/1.74 thf(func_def_3, type, dtree: $tType).
% 3.16/1.74 thf(func_def_4, type, set: ($tType) > $tType).
% 3.16/1.74 thf(func_def_5, type, t: $tType).
% 3.16/1.74 thf(func_def_6, type, n: $tType).
% 3.16/1.74 thf(func_def_7, type, itself: ($tType) > $tType).
% 3.16/1.74 thf(func_def_8, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.16/1.74 thf(func_def_9, type, ord: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.16/1.74 thf(func_def_10, type, top: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.16/1.74 thf(func_def_11, type, finite_finite: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.16/1.74 thf(func_def_12, type, semiring_char_0: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.16/1.74 thf(func_def_13, type, order_top: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.16/1.74 thf(func_def_14, type, bNF_Ca1785829860lChain: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X0 > X1) > $o)).
% 3.16/1.74 thf(func_def_15, type, bNF_fstOp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X1 > X2 > $o) > product_prod @ X0 @ X2 > product_prod @ X0 @ X1)).
% 3.16/1.74 thf(func_def_16, type, bNF_pick_middlep: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X1 > X2 > $o) > X0 > X2 > X1)).
% 3.16/1.74 thf(func_def_17, type, bNF_sndOp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X1 > X2 > $o) > product_prod @ X0 @ X2 > product_prod @ X1 @ X2)).
% 3.16/1.74 thf(func_def_18, type, bNF_Greatest_image2: !>[X0: $tType, X1: $tType, X2: $tType]:(set @ X0 > (X0 > X1) > (X0 > X2) > set @ product_prod @ X1 @ X2)).
% 3.16/1.74 thf(func_def_19, type, node: n > set @ sum_sum @ t @ dtree > dtree).
% 3.16/1.74 thf(func_def_20, type, cont: dtree > set @ sum_sum @ t @ dtree).
% 3.16/1.74 thf(func_def_21, type, corec: !>[X0: $tType]:((X0 > n) > (X0 > set @ sum_sum @ t @ sum_sum @ dtree @ X0) > X0 > dtree)).
% 3.16/1.74 thf(func_def_22, type, root: dtree > n).
% 3.16/1.74 thf(func_def_23, type, unfold: !>[X0: $tType]:((X0 > n) > (X0 > set @ sum_sum @ t @ X0) > X0 > dtree)).
% 3.16/1.74 thf(func_def_24, type, finite_finite2: !>[X0: $tType]:(set @ X0 > $o)).
% 3.16/1.74 thf(func_def_25, type, inj_on: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > $o)).
% 3.16/1.74 thf(func_def_26, type, swap: !>[X0: $tType, X1: $tType]:(X0 > X0 > (X0 > X1) > X0 > X1)).
% 3.16/1.74 thf(func_def_27, type, inf_inf: !>[X0: $tType]:(X0 > X0 > X0)).
% 3.16/1.74 thf(func_def_28, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.16/1.74 thf(func_def_29, type, top_top: !>[X0: $tType]:(X0)).
% 3.16/1.74 thf(func_def_30, type, parall1518086719_Nplus: n > n > n).
% 3.16/1.74 thf(func_def_31, type, parall1899940088le_par: product_prod @ dtree @ dtree > dtree).
% 3.16/1.74 thf(func_def_32, type, parall1914194347_par_c: product_prod @ dtree @ dtree > set @ sum_sum @ t @ product_prod @ dtree @ dtree).
% 3.16/1.74 thf(func_def_33, type, parall1914194362_par_r: product_prod @ dtree @ dtree > n).
% 3.16/1.74 thf(func_def_34, type, parall556292031_r_rel: product_prod @ dtree @ dtree > product_prod @ dtree @ dtree > $o).
% 3.16/1.74 thf(func_def_35, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 3.16/1.74 thf(func_def_36, type, product_curry: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X0 @ X1 > X2) > X0 > X1 > X2)).
% 3.16/1.74 thf(func_def_37, type, produc2004651681e_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.16/1.74 thf(func_def_38, type, product_map_prod: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1) > (X2 > X3) > product_prod @ X0 @ X2 > product_prod @ X1 @ X3)).
% 3.16/1.74 thf(func_def_39, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.16/1.74 thf(func_def_40, type, product_case_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 3.16/1.74 thf(func_def_41, type, product_fst: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 3.16/1.74 thf(func_def_42, type, product_snd: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 3.16/1.74 thf(func_def_43, type, product_swap: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > product_prod @ X1 @ X0)).
% 3.16/1.74 thf(func_def_44, type, product_scomp: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > product_prod @ X1 @ X2) > (X1 > X2 > X3) > X0 > X3)).
% 3.16/1.74 thf(func_def_45, type, type2: !>[X0: $tType]:(itself @ X0)).
% 3.16/1.74 thf(func_def_46, type, inv_image: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X1 > X0) > set @ product_prod @ X1 @ X1)).
% 3.16/1.74 thf(func_def_47, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 3.16/1.74 thf(func_def_48, type, pow: !>[X0: $tType]:(set @ X0 > set @ set @ X0)).
% 3.16/1.74 thf(func_def_49, type, image: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1)).
% 3.16/1.74 thf(func_def_50, type, vimage: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X1 > set @ X0)).
% 3.16/1.74 thf(func_def_51, type, accp: !>[X0: $tType]:((X0 > X0 > $o) > X0 > $o)).
% 3.16/1.74 thf(func_def_52, type, lex_prod: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ X1 @ X1 > set @ product_prod @ product_prod @ X0 @ X1 @ product_prod @ X0 @ X1)).
% 3.16/1.74 thf(func_def_53, type, same_fst: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X0 > set @ product_prod @ X1 @ X1) > set @ product_prod @ product_prod @ X0 @ X1 @ product_prod @ X0 @ X1)).
% 3.16/1.74 thf(func_def_54, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 3.16/1.74 thf(func_def_55, type, tr1: dtree).
% 3.16/1.74 thf(func_def_56, type, tr2: dtree).
% 3.16/1.74 thf(func_def_70, type, sK0: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.16/1.74 thf(func_def_71, type, sK1: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.16/1.74 thf(func_def_72, type, sK2: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.16/1.74 thf(func_def_73, type, sK3: !>[X0: $tType]:(set @ X0 > X0)).
% 3.16/1.74 thf(func_def_74, type, sK4: !>[X0: $tType, X1: $tType]:(X0 > (X1 > X0) > X1)).
% 3.16/1.74 thf(func_def_75, type, sK5: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 3.16/1.74 thf(func_def_76, type, sK6: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 3.16/1.74 thf(func_def_77, type, sK7: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.16/1.74 thf(func_def_78, type, sK8: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.16/1.74 thf(func_def_79, type, sK9: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.16/1.74 thf(func_def_80, type, sK10: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 3.16/1.74 thf(func_def_81, type, sK11: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.16/1.74 thf(func_def_82, type, sK12: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.16/1.74 thf(func_def_83, type, sK13: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.16/1.74 thf(func_def_84, type, sK14: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X0)).
% 3.16/1.74 thf(func_def_85, type, sK15: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X1)).
% 3.16/1.74 thf(func_def_86, type, sK16: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X2)).
% 3.16/1.74 thf(func_def_87, type, sK17: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X3)).
% 3.16/1.74 thf(func_def_88, type, sK18: product_prod @ dtree @ dtree > dtree).
% 3.16/1.74 thf(func_def_89, type, sK19: product_prod @ dtree @ dtree > dtree).
% 3.16/1.74 thf(func_def_90, type, sK20: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > X0) > set @ X0 > set @ X1)).
% 3.16/1.74 thf(func_def_91, type, sK21: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 3.16/1.74 thf(func_def_92, type, sK22: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 3.16/1.74 thf(func_def_93, type, sK23: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X0)).
% 3.16/1.74 thf(func_def_94, type, sK24: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X1)).
% 3.16/1.74 thf(func_def_95, type, sK25: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X2)).
% 3.16/1.74 thf(func_def_96, type, sK26: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X3)).
% 3.16/1.74 thf(func_def_97, type, sK27: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X4)).
% 3.16/1.74 thf(func_def_98, type, sK28: (product_prod @ dtree @ dtree > $o) > dtree).
% 3.16/1.74 thf(func_def_99, type, sK29: (product_prod @ dtree @ dtree > $o) > dtree).
% 3.16/1.74 thf(func_def_100, type, sK30: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 3.16/1.74 thf(func_def_101, type, sK31: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X6)).
% 3.16/1.74 thf(func_def_102, type, sK32: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 3.16/1.74 thf(func_def_103, type, sK33: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 3.16/1.74 thf(func_def_104, type, sK34: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 3.16/1.74 thf(func_def_105, type, sK35: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.16/1.74 thf(func_def_106, type, sK36: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.16/1.74 thf(func_def_107, type, sK37: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.16/1.74 thf(func_def_108, type, sK38: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X0)).
% 3.16/1.74 thf(func_def_109, type, sK39: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X1)).
% 3.16/1.74 thf(func_def_110, type, sK40: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X2)).
% 3.16/1.74 thf(func_def_111, type, sK41: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X3)).
% 3.16/1.74 thf(func_def_112, type, sK42: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X4)).
% 3.16/1.74 thf(func_def_113, type, sK43: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X5)).
% 3.16/1.74 thf(func_def_114, type, sK44: !>[X0: $tType, X1: $tType]:((X0 > X1) > X1)).
% 3.16/1.74 thf(func_def_115, type, sK45: !>[X0: $tType, X1: $tType]:(X1 > (X0 > X1) > X0)).
% 3.16/1.74 thf(func_def_116, type, sK46: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 3.16/1.74 thf(func_def_117, type, sK47: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 3.16/1.74 thf(func_def_118, type, sK48: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 3.16/1.74 thf(func_def_119, type, sK49: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 3.16/1.74 thf(func_def_120, type, sK50: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 3.16/1.74 thf(func_def_121, type, sK51: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 3.16/1.74 thf(func_def_122, type, sK52: n > product_prod @ dtree @ dtree > dtree).
% 3.16/1.74 thf(func_def_123, type, sK53: n > product_prod @ dtree @ dtree > dtree).
% 3.16/1.74 thf(func_def_124, type, sK54: !>[X0: $tType, X1: $tType]:(X0 > (X1 > X0) > X1)).
% 3.16/1.74 thf(func_def_125, type, sK55: !>[X0: $tType]:(set @ X0 > X0)).
% 3.16/1.74 thf(func_def_126, type, sK56: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 3.16/1.74 thf(func_def_127, type, sK57: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 3.16/1.74 thf(func_def_128, type, sK58: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X1 > X1)).
% 3.16/1.74 thf(func_def_129, type, sK59: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X1 > X1 > X0)).
% 3.16/1.74 thf(func_def_130, type, sK60: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 3.16/1.74 thf(func_def_131, type, sK61: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 3.16/1.74 thf(func_def_132, type, sK62: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X0 > X1) > X1)).
% 3.16/1.74 thf(func_def_133, type, sK63: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X0)).
% 3.16/1.74 thf(func_def_134, type, sK64: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X1)).
% 3.16/1.74 thf(func_def_135, type, sK65: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X2)).
% 3.16/1.74 thf(func_def_136, type, sK66: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 3.16/1.74 thf(func_def_137, type, sK67: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 3.16/1.74 thf(func_def_138, type, sK68: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X0)).
% 3.16/1.74 thf(func_def_139, type, sK69: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X1)).
% 3.16/1.74 thf(func_def_140, type, sK70: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X2)).
% 3.16/1.74 thf(func_def_141, type, sK71: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X3)).
% 3.16/1.74 thf(func_def_142, type, sK72: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X4)).
% 3.16/1.74 thf(func_def_143, type, sK73: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X5)).
% 3.16/1.74 thf(func_def_144, type, sK74: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X6)).
% 3.16/1.74 thf(func_def_145, type, sK75: !>[X0: $tType]:(X0)).
% 3.16/1.74 thf(func_def_147, type, ph77: !>[X0: $tType]:(X0)).
% 3.16/1.74 thf(f886,plain,(
% 3.16/1.74 $false),
% 3.16/1.74 inference(subsumption_resolution,[status(thm)],[f885,f865])).
% 3.16/1.74 thf(f865,plain,(
% 3.16/1.74 ( ! [X0 : dtree,X1 : dtree] : (((parall1914194362_par_r @ (product_Pair @ dtree @ dtree @ X0 @ X1)) = (parall1518086719_Nplus @ (root @ X0) @ (root @ X1)))) )),
% 3.16/1.74 inference(cnf_transformation,[status(thm)],[f658])).
% 3.16/1.74 thf(f658,plain,(
% 3.16/1.74 ! [X0 : dtree,X1 : dtree] : ((parall1914194362_par_r @ (product_Pair @ dtree @ dtree @ X0 @ X1)) = (parall1518086719_Nplus @ (root @ X0) @ (root @ X1)))),
% 3.16/1.74 inference(rectify,[status(thm)],[f6])).
% 3.16/1.74 thf(f6,axiom,(
% 3.16/1.74 ! [X14 : dtree,X15 : dtree] : ((parall1914194362_par_r @ (product_Pair @ dtree @ dtree @ X14 @ X15)) = (parall1518086719_Nplus @ (root @ X14) @ (root @ X15)))),
% 3.16/1.74 file('/export/starexec/sandbox/tmp/tmp.qb6mFvvipo/DTF2THF_20809.p',fact_5_par__r_Osimps)).
% 3.16/1.74 thf(f885,plain,(
% 3.16/1.74 ((parall1914194362_par_r @ (product_Pair @ dtree @ dtree @ tr1 @ tr2)) != (parall1518086719_Nplus @ (root @ tr1) @ (root @ tr2)))),
% 3.16/1.74 inference(superposition,[status(thm)],[f840,f866])).
% 3.16/1.74 thf(f866,plain,(
% 3.16/1.74 ( ! [X0 : $tType,X2 : X0 > set @ sum_sum @ t @ X0,X3 : X0,X1 : X0 > n] : (((X1 @ X3) = (root @ (unfold @ X0 @ X1 @ X2 @ X3)))) )),
% 3.16/1.74 inference(cnf_transformation,[status(thm)],[f657])).
% 3.16/1.74 thf(f657,plain,(
% 3.16/1.74 ! [X0 : $tType,X1 : X0 > n,X2 : X0 > set @ sum_sum @ t @ X0,X3 : X0] : ((X1 @ X3) = (root @ (unfold @ X0 @ X1 @ X2 @ X3)))),
% 3.16/1.74 inference(rectify,[status(thm)],[f9])).
% 3.16/1.74 thf(f9,axiom,(
% 3.16/1.74 ! [X0 : $tType,X18 : X0 > n,X19 : X0 > set @ sum_sum @ t @ X0,X8 : X0] : ((root @ (unfold @ X0 @ X18 @ X19 @ X8)) = (X18 @ X8))),
% 3.16/1.74 file('/export/starexec/sandbox/tmp/tmp.qb6mFvvipo/DTF2THF_20809.p',fact_8_unfold_I1_J)).
% 3.16/1.74 thf(f840,plain,(
% 3.16/1.74 ((root @ (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c @ (product_Pair @ dtree @ dtree @ tr1 @ tr2))) != (parall1518086719_Nplus @ (root @ tr1) @ (root @ tr2)))),
% 3.16/1.74 inference(cnf_transformation,[status(thm)],[f655])).
% 3.16/1.74 thf(f655,plain,(
% 3.16/1.74 ((root @ (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c @ (product_Pair @ dtree @ dtree @ tr1 @ tr2))) != (parall1518086719_Nplus @ (root @ tr1) @ (root @ tr2)))),
% 3.16/1.74 inference(flattening,[status(thm)],[f273])).
% 3.16/1.74 thf(f273,negated_conjecture,(
% 3.16/1.74 ~((root @ (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c @ (product_Pair @ dtree @ dtree @ tr1 @ tr2))) = (parall1518086719_Nplus @ (root @ tr1) @ (root @ tr2)))),
% 3.16/1.74 inference(negated_conjecture,[status(cth)],[f272])).
% 3.16/1.74 thf(f272,conjecture,(
% 3.16/1.74 ((root @ (unfold @ product_prod @ dtree @ dtree @ parall1914194362_par_r @ parall1914194347_par_c @ (product_Pair @ dtree @ dtree @ tr1 @ tr2))) = (parall1518086719_Nplus @ (root @ tr1) @ (root @ tr2)))),
% 3.16/1.74 file('/export/starexec/sandbox/tmp/tmp.qb6mFvvipo/DTF2THF_20809.p',conj_0)).
% 3.16/1.74 % SZS output end Proof for DTF2THF_20809
% 3.16/1.74 % (20913)------------------------------
% 3.16/1.74 % (20913)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.16/1.74 % (20913)Termination reason: Refutation
% 3.16/1.74
% 3.16/1.74 % (20913)Memory used [KB]: 6652
% 3.16/1.74 % (20913)Time elapsed: 0.033 s
% 3.16/1.74 % (20913)Instructions burned: 59 (million)
% 3.16/1.74 % (20913)------------------------------
% 3.16/1.74 % (20913)------------------------------
% 3.16/1.74 % (20907)Success in time 0.062 s
%------------------------------------------------------------------------------